Explaining and Justifying Equation Solving Steps - Algebra 1

When solving $-4x = 12$ , a student divides both sides by $-4$ to get $x = -3$ . A classmate argues this step is invalid because "you can't divide by a negative number in equations." How should the first student respond?
Answer: The step is valid because the division property of equality applies to all nonzero numbers, including negatives When you encounter negative coefficients in linear equations, remember that the division property of equality works with all nonzero numbers. This property states that if you divide both sides of an equation by the same nonzero number, the equation remains balanced. The first student's work is completely correct. Dividing both sides of $-4x = 12$ by $-4$ gives $x = \frac{12}{-4} = -3$ . You can verify this: substituting $x = -3$ back into the original equation yields $-4(-3) = 12$ , which simplifies to $12 = 12$ ✓. Let's examine why the other options are incorrect. Choice A is wrong because dividing by negative numbers doesn't create extraneous solutions—that typically happens with operations like squaring both sides or multiplying by expressions containing variables. Choice B confuses equations with inequalities; while dividing inequalities by negative numbers does flip the inequality sign, equations don't have directional signs to flip. Choice C suggests an unnecessary extra step—while multiplying both sides by $-1$ would work, it's not required since you can divide directly by the negative coefficient. Choice D correctly identifies that the division property of equality applies to all nonzero numbers, whether positive or negative. Study tip: Don't let negative coefficients intimidate you in linear equations. The same algebraic properties apply regardless of sign. However, do remember that inequalities are different—they require flipping the inequality symbol when multiplying or dividing by negative numbers.
Which property justifies the step from $5(x+2)=20$ to $x+2=4$ ? Step shown: $5(x+2)=20 \;\rightarrow\; x+2=4$
Answer: Division Property of Equality This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 5(x + 2) = 20 to x + 2 = 4: we divided both sides by 5. This is justified by the Division Property of Equality, which states that if we divide both sides by the same nonzero value, the equality is preserved. We can see this is valid because: left side (5(x + 2) ÷ 5 = x + 2) and right side (20 ÷ 5 = 4)—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Division Property of Equality because we divided both sides by 5 to eliminate the coefficient of the parenthetical expression. Choice A names the wrong property: it says Distributive Property, but we're actually dividing both sides by 5, which is the Division Property. It's easy to confuse the Distributive Property with the Division Property when parentheses are involved, but remember: the Distributive Property transforms one expression (like 5(x + 2) into 5x + 10), while the Division Property of Equality maintains balance between two sides of an equation by dividing both by the same value. Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!
Why is this step valid? Step shown: If $x-4=9$ , then $x-4+4=9+4$ . Which property of equality justifies this step?
Answer: Addition Property of Equality This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x - 4 = 9 to x - 4 + 4 = 9 + 4: we added 4 to both sides. This is justified by the Addition Property of Equality, which states that if we add the same value to both sides of an equation, the equality is preserved. We can see this is valid because we're adding the exact same value (4) to both the left side and the right side—this maintains the balance of the equation! Choice A correctly identifies the property as the Addition Property of Equality because we're adding 4 to both sides to isolate x. Choice D mentions the Symmetric Property of Equality, which states that if a = b, then b = a (we can flip the sides). But that's not what's happening here—we're not flipping sides, we're adding 4 to both sides. The Symmetric Property is about reversing the order of an equation, while the Addition Property is about adding the same value to maintain equality. A complete justification has three parts: (1) What you did ('added 4'), (2) To where ('to both sides'), (3) Which property justifies it ('Addition Property of Equality'). Example: 'Added 4 to both sides using the Addition Property of Equality.' This format ensures you've covered all the bases!
A student factors and then solves: 1) $(x-2)(x+5)=0$ 2) $x-2=0$ or $x+5=0$ What property allows the student to go from line 1 to line 2?
Answer: Zero Product Property This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '(x-2)(x+5)=0 → x-2=0 or x+5=0' is justified by the Zero Product Property: this property states that if a product is zero, then at least one factor must be zero. In this step, we set each factor equal to zero, which allows solving for x. This is different from Distributive Property because we're dealing with the condition for zero product, not expansion. Choice C correctly identifies the property as the Zero Product Property because it enables splitting the equation into two simpler equations. Choice A confuses the Zero Product Property with the Distributive Property: the Distributive Property transforms one expression (like (x-2)(x+5) into x² + 3x - 10), while the Zero Product Property maintains the factored form and sets factors to zero. They're both important but serve different purposes in solving! Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!
Fill in the reasoning for the missing step. 1) $\sqrt{x+3}=5$ 2) $x+3=25$ 3) $x=22$ Why is the step from line 1 to line 2 valid?
Answer: Power Property of Equality: square both sides of the equation. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '√(x+3) = 5 → x+3 = 25' is justified by the Power Property of Equality: this property states that if a = b and n is a positive integer, then $a^n$ = $b^n$ (here, squaring both sides with n=2). In this step, we raised both sides to the power of 2, which maintains equality provided we check for extraneous solutions later. This is different from the Multiplication Property because we're exponentiating, not just multiplying by a constant. Choice C correctly identifies the justification as the Power Property of Equality because squaring both sides eliminates the square root while preserving potential solutions. Choice A names the wrong property: it says Multiplication Property, but we're actually raising to a power, which is the Power Property. It's easy to confuse Multiplication with Power, but remember: Multiplication is about multiplying by the same nonzero value, while Power is about raising both sides to the same exponent. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four! When justifying, identify which one applies and always mention 'both sides.'
Construct a viable argument showing that if the equation $\dfrac{1}{x}+2=3$ has a solution, it must satisfy $x=1$ . Which justification correctly supports the step $\dfrac{1}{x}=1 \rightarrow 1=x$ ?
Answer: Symmetric Property of Equality: if $a=b$ , then $b=a$ . This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '1/x = 1 → 1 = x' is justified by the Symmetric Property of Equality: this property states that if a = b, then b = a. In this step, we swapped the sides of the equation, which maintains equality. This is different from the Addition Property because we're reordering the equality, not adding anything. Choice A correctly identifies the justification as the Symmetric Property of Equality because it allows us to rewrite the equation with x on the left for clarity, preserving the solution. Choice B names the wrong property: it says Division Property, but we're actually swapping sides, which is the Symmetric Property. It's easy to confuse Division with Symmetric here since 1/x=1 implies x=1 by taking reciprocals, but the specific step shown is just swapping. A complete justification has three parts: (1) What you did ('swapped sides'), (2) To where ('both sides of the equality'), (3) Which property justifies it ('Symmetric Property of Equality'). This format ensures you've covered all the bases!
A student is solving the equation $3(x-4)+2=11$ and writes the steps below. 1) $3(x-4)+2=11$ 2) $3x-12+2=11$ 3) $3x-10=11$ Which statement correctly justifies the step from line 1 to line 2?
Answer: Distributive Property: distribute $3$ across $(x-4)$ to get $3x-12$ . This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The Distributive Property (a(b + c) = ab + ac) is different from the properties of equality: it's about transforming one expression into an equivalent form, not about maintaining equality between two sides. But it's used frequently in equation solving—like expanding 2(x - 3) to 2x - 6—and needs to be distinguished from the equality properties that justify working with both sides. Looking at the transformation from 3(x-4)+2=11 to 3x-12+2=11: we distributed the 3 over (x-4) on the left side, which is justified by the Distributive Property, which states that a(b - c) = ab - ac. We can see this is valid because it only transforms the expression within one side without affecting the equality. Choice B correctly identifies the property as the Distributive Property because it specifically names the action of distributing 3 across (x-4) to get 3x-12, which is exactly what happened on the left side. Choice A names the wrong property: it says Addition Property of Equality, but we're actually distributing a multiplier, which is the Distributive Property. It's easy to confuse Addition Property with Distributive Property, but remember: Addition Property is about adding the same value to both sides, while Distributive Property is about expanding expressions. Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!
What justification is used in going from line 2 to line 3? 1) $4x+7=3x-5$ 2) $x+7=-5$ 3) $x=-12$
Answer: Subtraction Property of Equality: subtract $7$ from both sides. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x + 7 = -5 to x = -12: we subtracted 7 from both sides. This is justified by the Subtraction Property of Equality, which states that if a = b, then a - c = b - c for any c. We can see this is valid because left side x + 7 - 7 = x and right side -5 - 7 = -12—same operation applied to both sides, so equality is preserved! Choice A correctly identifies the property as the Subtraction Property of Equality because subtracting 7 from both sides isolates x. Choice B names the wrong property: it says Addition Property, but we're actually subtracting 7, which is the Subtraction Property. It's easy to confuse Addition with Subtraction, but remember: Addition is about adding the same to both sides, while Subtraction is about subtracting the same. A complete justification has three parts: (1) What you did ('subtracted 7'), (2) To where ('from both sides'), (3) Which property justifies it ('Subtraction Property of Equality'). This format ensures you've covered all the bases!
Solve $2(x-3)=10$ and justify each step using properties of equality and algebraic properties. Which sequence of justifications correctly matches a valid solution path? Step 1: $2(x-3)=10 \rightarrow 2x-6=10$ Step 2: $2x-6=10 \rightarrow 2x=16$ Step 3: $2x=16 \rightarrow x=8$
Answer: Step 1 Distributive Property; Step 2 Addition Property of Equality; Step 3 Division Property of Equality. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The Distributive Property (a(b + c) = ab + ac) is different from the properties of equality: it's about transforming one expression into an equivalent form, not about maintaining equality between two sides. But it's used frequently in equation solving—like expanding 2(x - 3) to 2x - 6—and needs to be distinguished from the equality properties that justify working with both sides. Let's solve 2(x-3)=10 with full justification: Starting equation: 2(x-3)=10 (Given). Step 1: Distribute 2 → 2x-6=10 (Justification: Distributive Property because 2*(x-3)=2x-6 expands the left side equivalently). Step 2: Add 6 to both sides → 2x=16 (Justification: Addition Property of Equality because adding 6 to both sides isolates the term with x). Step 3: Divide both sides by 2 → x=8 (Justification: Division Property of Equality because dividing both sides by 2 (nonzero) isolates x). We've constructed a valid argument showing that IF the equation has a solution, it must be x=8. Choice A correctly identifies the justifications as Step 1 Distributive Property, Step 2 Addition Property of Equality, Step 3 Division Property of Equality because each matches the operation: distributing in Step 1, adding to both sides in Step 2, and dividing both sides in Step 3. Choice D identifies the wrong properties: it says Step 1 Commutative, but we're actually distributing, which is the Distributive Property. It's easy to confuse Commutative with Distributive, but remember: Commutative is about order (a + b = b + a), while Distributive is about multiplying over addition. Don't confuse properties: The Distributive Property (a(b+c) = ab + ac) transforms one expression. The properties of equality transform both sides of an equation. Other algebraic properties like Commutative (a + b = b + a) and Associative ((a + b) + c = a + (b + c)) also transform expressions. Properties of equality are specifically about maintaining balance in equations—they're the 'do to both sides' rules!
What property allows you to go from $\dfrac{x}{5}=7$ to $x=35$ ? Step shown: $\dfrac{x}{5}=7 \;\rightarrow\; x=35$
Answer: Multiplication Property of Equality This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from x/5 = 7 to x = 35: we multiplied both sides by 5. This is justified by the Multiplication Property of Equality, which states that if we multiply both sides by the same nonzero value, the equality is preserved. We can see this is valid because: left side (x/5 × 5 = x) and right side (7 × 5 = 35)—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Multiplication Property of Equality because we multiplied both sides by 5 to eliminate the fraction and isolate x. Choice A names the wrong property: it says Division Property, but we're actually multiplying both sides by 5, which is the Multiplication Property. It's easy to think 'we're dealing with division in x/5, so use Division Property,' but remember: to undo division by 5, we MULTIPLY by 5. The Multiplication Property justifies this step! The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four! When justifying, identify which one applies and always mention 'both sides.'
Which property justifies the step from $\frac{x-1}{4}=3$ to $x-1=12$ ?
Answer: Multiplication Property of Equality This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from $\frac{x-1}{4}=3$ to $x-1=12$ : we multiplied both sides by 4. This is justified by the Multiplication Property of Equality, which states that if we multiply both sides by the same nonzero value, the equality is preserved. We can see this is valid because left side: $4 \cdot \frac{x-1}{4} = x-1$ and right side: $4 \cdot 3 = 12$ —same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Multiplication Property of Equality because we multiplied both sides by 4 to clear the denominator. Choice A (Division) would mean dividing both sides, choice C (Subtraction) would mean subtracting from both sides, and choice D (Distributive) is about transforming expressions, not maintaining equality between sides. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four!
Identify the error in the student work shown for solving $4x-9=3$ . Student work: 1) $4x-9=3$ 2) $4x=12$ 3) $x=12-4$ 4) $x=8$ Which statement best identifies the first incorrect step and why it is invalid?
Answer: Step 3 is incorrect: dividing both sides by $4$ should give $x=\dfrac{12}{4}$ , not $x=12-4$ . This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. A common error in equation solving is applying operations to only one side of the equation: if you subtract 5 from the left side, you MUST subtract 5 from the right side too. This is what the properties of equality guarantee—we do the SAME thing to BOTH sides to maintain the equality. Forgetting this 'both sides' rule is how equations get broken! Let's examine the student work: Step 1: 4x - 9 = 3 (given), Step 2: 4x = 12 (correct—added 9 to both sides), Step 3: x = 12 - 4 (ERROR!), Step 4: x = 8. The error occurs at Step 3: the student wrote x = 12 - 4 instead of dividing both sides by 4. The student appears to have subtracted 4 from 12, which violates the Division Property of Equality because to isolate x from 4x = 12, we must divide BOTH sides by 4, giving x = 12/4 = 3. The correct step would be 4x = 12 → x = 3, applying the Division Property of Equality. Choice B correctly identifies the error as occurring in Step 3 and explains that dividing both sides by 4 should give x = 12/4, not x = 12 - 4, because the Division Property requires dividing both sides by the same value. Choice A incorrectly claims Step 2 has an error, but Step 2 is actually correct: the student properly added 9 to both sides using the Addition Property of Equality. Step 2 transforms 4x - 9 = 3 to 4x = 12, which is valid. The error is in Step 3, not Step 2! For error analysis, go through the student work line by line asking: (1) Is each step justified by a property? (2) Was the same operation applied to both sides? (3) Was arithmetic correct? The error will be where one of these fails. Then explain: 'At Step 3, writing x = 12 - 4 instead of x = 12/4 violates the Division Property because we must divide both sides by 4.' Pinpointing the exact step and naming the violated property is key!
Which property justifies the step from $3x+5=14$ to $3x=9$ ? Step shown: $3x+5=14 \;\rightarrow\; 3x=9$
Answer: Subtraction Property of Equality This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 3x + 5 = 14 to 3x = 9: we subtracted 5 from both sides. This is justified by the Subtraction Property of Equality, which states that if we subtract the same value from both sides of an equation, the equality is preserved. We can see this is valid because: left side (3x + 5 - 5 = 3x) and right side (14 - 5 = 9)—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Subtraction Property of Equality because we subtracted 5 from both sides to isolate the term with x. Choice A names the wrong property: it says Addition Property, but we're actually subtracting 5 from both sides, which is the Subtraction Property. It's easy to confuse Addition with Subtraction when thinking about 'undoing' operations, but remember: Subtraction Property is about subtracting the same value from both sides, while Addition Property is about adding the same value to both sides. A complete justification has three parts: (1) What you did ('subtracted 5'), (2) To where ('from both sides'), (3) Which property justifies it ('Subtraction Property of Equality'). Example: 'Subtracted 5 from both sides using the Subtraction Property of Equality.' This format ensures you've covered all the bases!
A student solving $5x - 3 = 2x + 9$ writes the next step as $3x - 3 = 9$ . What mathematical reasoning supports this transformation?
Answer: Subtracting $2x$ from both sides maintains equality because equal quantities subtracted from equal quantities remain equal The subtraction property of equality (or addition property with $-2x$ ) justifies this step. Starting from $5x - 3 = 2x + 9$ , subtracting $2x$ from both sides gives $(5x - 2x) - 3 = (2x - 2x) + 9$ , which simplifies to $3x - 3 = 9$ . Choice B describes what happens but not why it's valid. Choices C and D incorrectly reference reflexive and symmetric properties, which don't apply to this algebraic manipulation.
A student solving $x^2 - 9 = 0$ factors to get $(x-3)(x+3) = 0$ and then writes $x = 3$ or $x = -3$ . What property allows the conclusion that at least one factor must equal zero?
Answer: The zero product property states that if $ab = 0$ , then $a = 0$ or $b = 0$ (or both) The zero product property (or null factor law) states that if the product of two or more factors equals zero, then at least one factor must equal zero. This property is fundamental to solving quadratic equations by factoring. Here, since $(x-3)(x+3) = 0$ , either $x-3 = 0$ or $x+3 = 0$ (or both), giving $x = 3$ or $x = -3$ . Choice A misapplies the distributive property. Choices C and D reference non-standard or vague properties.
Explain which property of equality justifies the step $5x-7=18 \rightarrow 5x=25$ .
Answer: Addition Property of Equality: add $7$ to both sides. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. The properties of equality are the rules that allow us to transform equations while maintaining balance: the Addition Property says we can add the same thing to both sides, the Subtraction Property says we can subtract the same thing from both sides, the Multiplication Property says we can multiply both sides by the same nonzero value, and the Division Property says we can divide both sides by the same nonzero value. Each step in solving must be justified by one of these properties! Looking at the transformation from 5x - 7 = 18 to 5x = 25: we added 7 to both sides, which is justified by the Addition Property of Equality, which states we can add the same value to both sides. We can see this is valid because left side: 5x - 7 + 7 = 5x, and right side: 18 + 7 = 25—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Addition Property of Equality because it specifies adding 7 to both sides, which isolates the term with x. Choice A names the wrong property: it says Subtraction Property, but we're actually adding 7, which is the Addition Property. It's easy to confuse Subtraction Property with Addition Property, but remember: Subtraction Property is about subtracting the same value from both sides, while Addition Property is about adding. The four main properties of equality to memorize: (1) Addition Property—add same to both sides, (2) Subtraction Property—subtract same from both sides, (3) Multiplication Property—multiply both sides by same nonzero value, (4) Division Property—divide both sides by same nonzero value. Almost every equation-solving step uses one of these four! When justifying, identify which one applies and always mention 'both sides.'
Two students solve $2(x-3)=4x+2$ . Student A: 1) $2(x-3)=4x+2$ 2) $2x-6=4x+2$ 3) $-6=2x+2$ 4) $-8=2x$ 5) $x=-4$ Student B: 1) $2(x-3)=4x+2$ 2) $2x-6=4x+2$ 3) $-2x-6=2$ 4) $-2x=8$ 5) $x=-4$ Which statement correctly explains why line 3 of Student A ( $2x-6=4x+2 \rightarrow -6=2x+2$ ) is valid?
Answer: Subtraction Property of Equality: subtract $2x$ from both sides and simplify. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. The step '2x - 6 = 4x + 2 → -6 = 2x + 2' is justified by the Subtraction Property: this property states that we can subtract the same value from both sides. In this step, we subtracted 2x from both sides, which maintains equality. This is different from adding because we're removing 2x terms. Choice C correctly identifies the property as the Subtraction Property of Equality because it specifies subtracting 2x from both sides and simplifying, which accurately describes the transformation. Choice A names the wrong property: it says subtract 2x from both sides, but omits the simplification note, though more importantly, Choice C includes it for clarity. It's easy to confuse similar descriptions, but remember: Subtraction Property is about subtracting the same value from both sides. A complete justification has three parts: (1) What you did ('subtracted 5,' 'divided by 3'), (2) To where ('from both sides,' 'both sides by'), (3) Which property justifies it ('Subtraction Property of Equality'). Example: 'Subtracted 5 from both sides using the Subtraction Property of Equality.' This format ensures you've covered all the bases!
Explain each step in solving $2(x-3)=10$ by naming the property used in each step. Work shown: 1) $2(x-3)=10$ 2) $2x-6=10$ 3) $2x=16$ 4) $x=8$ Which statement correctly explains step 2 ( $2x-6=10 \rightarrow 2x=16$ )?
Answer: Addition Property of Equality: add $6$ to both sides This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Looking at the transformation from 2x - 6 = 10 to 2x = 16: we added 6 to both sides. This is justified by the Addition Property of Equality, which states that if we add the same value to both sides of an equation, the equality is preserved. We can see this is valid because: left side (2x - 6 + 6 = 2x) and right side (10 + 6 = 16)—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Addition Property of Equality and specifies that we add 6 to both sides because that's exactly what happened to transform 2x - 6 = 10 into 2x = 16. Choice A says Subtraction Property and subtract 6, but we're actually adding 6 to both sides to eliminate the -6 on the left side. It's easy to think 'we're getting rid of -6 so we subtract,' but remember: to eliminate -6, we ADD 6 (because -6 + 6 = 0). The Addition Property is what justifies this step! When checking if a step is valid, ask yourself: 'Did I do the EXACT SAME thing to BOTH sides of the equation?' If yes, and you used one of the properties of equality, the step is valid. If you only operated on one side, or did different things to each side, the step breaks the equation and is invalid. The 'both sides' check catches most errors!
Construct a viable argument showing that if $7-2x=1$ has a solution, it must be $x=3$ . Which sequence correctly uses properties of equality?
Answer: $7-2x=1$ → $-2x=-6$ (Subtraction Property of Equality: subtract $7$ from both sides) → $x=3$ (Division Property of Equality: divide both sides by $-2$ ) This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Let's solve $7-2x=1$ with full justification: Starting equation: $7-2x=1$ (Given). Step 1: Subtract 7 from both sides → $-2x=-6$ (Justification: Subtraction Property of Equality because we subtracted 7 from both sides; left side: $7-2x-7=-2x$ , right side: $1-7=-6$ ). Step 2: Divide both sides by -2 → $x=3$ (Justification: Division Property of Equality because we divided both sides by -2; left side: $\frac{-2x}{-2}=x$ , right side: $\frac{-6}{-2}=3$ ). We've constructed a valid argument showing that IF the equation has a solution, it must be 3. Choice A correctly identifies both steps and properties: Subtraction Property for subtracting 7 from both sides to get $-2x=-6$ , then Division Property for dividing both sides by -2 to get $x=3$ . Choice B has arithmetic errors (adding 7 to 1 gives 8, not 6), choice C also has arithmetic errors, and choice D incorrectly invokes the Reflexive Property and Zero Product Property which don't apply here. A complete justification has three parts: (1) What you did ('subtracted 7,' 'divided by -2'), (2) To where ('from both sides,' 'both sides by'), (3) Which property justifies it ('Subtraction Property of Equality'). This format ensures you've covered all the bases!
Identify the error in the student work shown for solving $\sqrt{x+3}=5$ . Student work: 1) $\sqrt{x+3}=5$ 2) $x+3=25$ 3) $x=22$ 4) $\sqrt{22+3}=5$ so the solution is $x=22$ Which statement best explains the error?
Answer: There is no error; each step follows from properties of equality and the check is correct. This question tests your understanding of the mathematical reasoning behind equation solving—specifically, which properties of equality justify each step and ensure that each transformation preserves the solution. When we solve an equation, we're essentially building a logical argument: 'Assume the equation has a solution x. Then [applying properties step by step], we find x = [value]. Therefore, IF the equation has a solution, it MUST be this value.' Each step must follow logically from the previous one using a valid property—this is what makes our solution mathematically sound. Let's examine the student work: Step 1: √(x + 3) = 5 (given), Step 2: x + 3 = 25 (squared both sides), Step 3: x = 22 (subtracted 3 from both sides), Step 4: √(22 + 3) = √25 = 5 ✓ (verification). Each step is valid! Step 2 uses the property that if a = b, then a² = b² (squaring both sides maintains equality for positive values). Step 3 uses the Subtraction Property of Equality. The check in Step 4 confirms that x = 22 satisfies the original equation. Choice D correctly states there is no error because each step follows from properties of equality and the check verifies the solution is correct. Choice C claims the check √(22 + 3) = √25 = 5 is false, but this is incorrect: √25 = 5 is true! The student correctly verified that when x = 22, the left side of the original equation becomes √(22 + 3) = √25 = 5, which equals the right side. The solution and verification are both correct. When checking if a step is valid, ask yourself: 'Did I do the EXACT SAME thing to BOTH sides of the equation?' If yes, and you used one of the properties of equality, the step is valid. Squaring both sides is a valid operation (though we must be careful about extraneous solutions with radicals), and the student's work here is completely correct!

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