Algebra Quiz: Explaining And Justifying Equation Solving Steps

What this quiz covers

This quiz focuses on Explaining And Justifying Equation Solving Steps, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Step 1: $2(3x-1)=10$ → $6x-2=10$ (Distributive Property) Step 2: $6x-2=10$ → $6x=12$ (Addition Property of Equality) Step 3: $6x=12$ → $x=2$ (Division Property of Equality) (correct answer)
2. Step 1: $2(3x-1)=10$ → $3x-1=5$ (Division Property of Equality) Step 2: $3x-1=5$ → $3x=4$ (Subtraction Property of Equality) Step 3: $3x=4$ → $x=\frac{4}{3}$ (Multiplication Property of Equality)
3. Step 1: $2(3x-1)=10$ → $6x-1=10$ (Distributive Property) Step 2: $6x-1=10$ → $6x=9$ (Addition Property of Equality) Step 3: $6x=9$ → $x=\frac{3}{2}$ (Division Property of Equality)
4. Step 1: $2(3x-1)=10$ → $6x-2=10$ (Commutative Property) Step 2: $6x-2=10$ → $x-2=10$ (Division Property of Equality) Step 3: $x-2=10$ → $x=12$ (Addition Property of Equality)

Explanation: 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 $2(3x-1)=10$ with full justification: Starting equation: $2(3x-1)=10$ (Given). Step 1: Apply Distributive Property to expand $2(3x-1)$ → $6x-2=10$ (Justification: Distributive Property because $2(3x-1) = 2(3x) + 2(-1) = 6x - 2$). Step 2: Add 2 to both sides → $6x=12$ (Justification: Addition Property of Equality because we added 2 to both sides). Step 3: Divide both sides by 6 → $x=2$ (Justification: Division Property of Equality because we divided both sides by 6). We've constructed a valid argument showing that IF the equation has a solution, it must be 2. Choice A correctly identifies all steps and properties: Distributive Property for expanding, Addition Property for adding 2 to both sides, and Division Property for dividing both sides by 6. Choice B has arithmetic errors in step 2 (should add 1, not subtract), choice C distributes incorrectly ($2(3x-1) ≠ 6x-1$), and choice D misuses the Commutative Property and makes nonsensical steps. 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'). This format ensures you've covered all the bases!

Question 2

1. Multiplication Property of Equality
2. Subtraction Property of Equality
3. Addition Property of Equality (correct answer)
4. Symmetric Property of Equality

Explanation: 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!

Question 3

1. Line 2 is incorrect because $x^2-5x+6$ does not factor.
2. Line 3 is incorrect: by the Zero Product Property, $x-2=0$ or $x-3=0$. (correct answer)
3. Line 4 is incorrect: you should divide both sides by $x$ to solve.
4. There is no error; $x=-1$ and $x=1$ are the correct solutions.

Explanation: 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: 1) x² - 5x + 6 = 0, 2) (x-2)(x-3)=0, 3) x-2=-3 or x-3=-2, 4) x=-1 or x=1. The error occurs at Step 3: the student incorrectly set the factors equal to the negatives of the other factors, which violates the Zero Product Property because after factoring, you must set each factor equal to zero: (x-2)=0 or (x-3)=0, leading to x=2 or x=3. The correct step would be applying the Zero Product Property properly to get x=2 or x=3. Choice B correctly identifies the error as the incorrect application of the Zero Product Property in line 3 because the student did not set each factor to zero, resulting in wrong solutions. Choice D says there is no error, but that's incorrect: while the factoring in line 2 is right, line 3 misapplies the property needed to solve. Error analysis requires careful checking of each step! 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 [n], [what they did wrong] violates [property] because [reason].' Pinpointing the exact step and naming the violated property is key!

Question 4

1. Subtraction Property of Equality: subtract $7$ from both sides. (correct answer)
2. Multiplication Property of Equality: multiply both sides by $-2$.
3. Division Property of Equality: divide both sides by $7$.
4. Addition Property of Equality: add $7$ to both sides.

Explanation: 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 7 - 2x = 15 to -2x = 8: we subtracted 7 from both sides, which is justified by the Subtraction Property of Equality, which states we can subtract the same value from both sides. We can see this is valid because left side: 7 - 2x - 7 = -2x, and right side: 15 - 7 = 8—same operation applied to both sides, so equality is preserved! Choice B correctly identifies the property as the Subtraction Property of Equality because it specifies subtracting 7 from both sides, isolating the term with x. Choice A 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 Property is about adding the same value to both sides, while Subtraction is about subtracting. 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.'

Question 5

1. Line 2 is correct by the Division Property of Equality because the $x$ cancels automatically.
2. The student should multiply both sides by $x$ to get $2=6x$; line 2 incorrectly removed $x$ without performing the same operation on both sides. (correct answer)
3. The student should add $x$ to both sides to clear the denominator; line 2 should be $\frac{2}{x}+x=6+x$.
4. The error is in line 3 only; $x=1$ should be $x=3$ by the Addition Property of Equality.

Explanation: 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: Line 1: $\frac{2}{x}=6$, Line 2: $2=6$, Line 3: $x=1$. The error occurs at Step 2: the student appears to have just 'removed' the $x$ from the denominator without performing any operation on both sides. To solve $\frac{2}{x}=6$ correctly, we should multiply both sides by $x$ to get $2=6x$ (Multiplication Property of Equality), then divide both sides by 6 to get $x=\frac{1}{3}$ (Division Property of Equality). The student's line 2 makes no mathematical sense—you can't just make a variable disappear! Choice B correctly identifies that the student should multiply both sides by $x$ to get $2=6x$, and that line 2 incorrectly removed $x$ without performing the same operation on both sides. Choice A wrongly claims the step is correct, choice C suggests adding $x$ (which wouldn't help), and choice D points to the wrong line. 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!

Question 6

1. Line 2 is invalid because the student multiplied only the left side by $3$ instead of multiplying both sides by $3$ (Multiplication Property of Equality). (correct answer)
2. Line 3 is invalid because subtracting $4$ from both sides violates the Subtraction Property of Equality.
3. Line 2 is valid by the Distributive Property, since $\frac{x}{3}+4$ distributes to $x+4$.
4. There is no error; the solution $x=6$ is correct because $6/3+4=10$.

Explanation: 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: Line 1: $\frac{x}{3}+4=10$, Line 2: $x+4=10$, Line 3: $x=6$. The error occurs at Step 2: to go from $\frac{x}{3}+4=10$ to $x+4=10$, the student appears to have multiplied only the $\frac{x}{3}$ term by 3, getting $x$, but didn't multiply the entire left side by 3. The correct step would be to multiply BOTH SIDES by 3: $3(\frac{x}{3}+4)=3(10)$, which gives $x+12=30$, applying the Multiplication Property of Equality correctly. Choice A correctly identifies the error as multiplying only part of the left side by 3 instead of multiplying both entire sides by 3, violating the Multiplication Property of Equality. Choice B incorrectly points to line 3, choice C misunderstands the Distributive Property, and choice D claims there's no error when there clearly is one. 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!

Question 7

1. Zero Product Property: if $(x-2)(x-3)=0$, then $x-2=0$ or $x-3=0$ (correct answer)
2. Distributive Property: if $(x-2)(x-3)=0$, then $x^2-5x+6=0$
3. Addition Property of Equality: add $2$ to both sides to get $x=2$ or $x=3$
4. Division Property of Equality: divide both sides by $(x-2)$ to get $x-3=0$

Explanation: 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 Student A's step from (x - 2)(x - 3) = 0 to x - 2 = 0 or x - 3 = 0: this uses the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must equal zero. This is a special property that applies when we have a product equal to zero—it's different from the properties of equality but equally important in equation solving. We can apply it here because we have two factors multiplied together equaling zero. Choice A correctly identifies the Zero Product Property and explains it properly: if (x - 2)(x - 3) = 0, then either x - 2 = 0 or x - 3 = 0 (or both), which leads to the solutions x = 2 or x = 3. Choice D suggests using the Division Property to divide by (x - 2), but this is dangerous and incorrect! We cannot divide both sides by (x - 2) because it might equal zero (when x = 2). Dividing by zero is undefined and would lose the x = 2 solution. The Zero Product Property is the correct approach for equations where a product equals zero. A complete justification has three parts: (1) What you did ('applied Zero Product Property'), (2) To what ('to the factored form'), (3) Which property justifies it ('Zero Product Property: if ab = 0, then a = 0 or b = 0'). This property is special because it only works when the product equals zero—it wouldn't apply if we had (x - 2)(x - 3) = 5, for example!

Question 8

1. $7-2x=1$ → $-2x=-6$ (Subtraction Property of Equality: subtract $7$ from both sides) → $x=3$ (Division Property of Equality: divide both sides by $-2$) (correct answer)
2. $7-2x=1$ → $2x=6$ (Addition Property of Equality: add $7$ to both sides) → $x=3$ (Multiplication Property of Equality: multiply both sides by $2$)
3. $7-2x=1$ → $-2x=8$ (Addition Property of Equality: add $1$ to both sides) → $x=-4$ (Division Property of Equality: divide both sides by $2$)
4. $7-2x=1$ → $7=1+2x$ (Reflexive Property of Equality) → $x=3$ (Zero Product Property)

Explanation: 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!

Question 9

1. Subtraction Property of Equality: subtract $7$ from both sides. (correct answer)
2. Addition Property of Equality: add $7$ to both sides.
3. Division Property of Equality: divide both sides by $7$.
4. Distributive Property: distribute $x$ across $(7)$.

Explanation: 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!

Question 10

1. Distributive Property: distribute $5$ across $x+9$.
2. Multiplication Property of Equality: multiply both sides by $9$.
3. Subtraction Property of Equality: subtract $9$ from both sides. (correct answer)
4. Addition Property of Equality: add $9$ to both sides.

Explanation: 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 + 9 = 24 to 5x = 15: we subtracted 9 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 5x + 9 - 9 = 5x and right side 24 - 9 = 15—same operation applied to both sides, so equality is preserved! Choice A correctly identifies the property as the Subtraction Property of Equality because subtracting 9 from both sides isolates the term with x while keeping the equation equivalent. Choice B names the wrong property: it says Addition Property, but we're actually subtracting 9, 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 from both sides. A complete justification has three parts: (1) What you did ('subtracted 9'), (2) To where ('from both sides'), (3) Which property justifies it ('Subtraction Property of Equality'). This format ensures you've covered all the bases!

Question 11

1. Line 3 is incorrect: by the Zero Product Property, $x-2=0$ or $x-3=0$. (correct answer)
2. Line 4 is incorrect: you should divide both sides by $x$ to solve.
3. There is no error; $x=-1$ and $x=1$ are the correct solutions.
4. Line 2 is incorrect because $x^2-5x+6$ does not factor.

Explanation: 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: 1) x² - 5x + 6 = 0, 2) (x-2)(x-3)=0, 3) x-2=-3 or x-3=-2, 4) x=-1 or x=1. The error occurs at Step 3: the student incorrectly set the factors equal to the negatives of the other factors, which violates the Zero Product Property because after factoring, you must set each factor equal to zero: (x-2)=0 or (x-3)=0, leading to x=2 or x=3. The correct step would be applying the Zero Product Property properly to get x=2 or x=3. Choice B correctly identifies the error as the incorrect application of the Zero Product Property in line 3 because the student did not set each factor to zero, resulting in wrong solutions. Choice D says there is no error, but that's incorrect: while the factoring in line 2 is right, line 3 misapplies the property needed to solve. Error analysis requires careful checking of each step! 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 [n], [what they did wrong] violates [property] because [reason].' Pinpointing the exact step and naming the violated property is key!

Question 12

1. Distributive Property: distribute $3$ across $(x-4)$.
2. Addition Property of Equality: add $10$ to both sides.
3. Subtraction Property of Equality: subtract $2x$ from both sides. (correct answer)
4. Division Property of Equality: divide both sides by $3$.

Explanation: 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 - 10 = 2x - 7 to x - 10 = -7: we subtracted 2x 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 3x - 2x - 10 = x - 10 and right side 2x - 2x - 7 = -7—same operation applied to both sides, so equality is preserved! Choice C correctly identifies the property as the Subtraction Property of Equality because subtracting 2x from both sides isolates the variable term while preserving the equation's balance. Choice A names the wrong property: it says Distributive Property, but we're actually subtracting terms from both sides, which is the Subtraction Property. It's easy to confuse Distributive with Subtraction, but remember: Distributive is about expanding expressions like 3(x-4), while Subtraction is about doing the same subtraction to both 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! When justifying, identify which one applies and always mention 'both sides.'

Question 13

1. Division Property of Equality
2. Subtraction Property of Equality (correct answer)
3. Addition Property of Equality
4. Distributive Property

Explanation: 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!

Question 14

1. Distributive Property
2. Addition Property of Equality
3. Reflexive Property of Equality
4. Zero Product Property (correct answer)

Explanation: 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!

Question 15

1. Division Property of Equality
2. Multiplication Property of Equality (correct answer)
3. Addition Property of Equality
4. Subtraction Property of Equality

Explanation: 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.'

Question 16

1. Multiplying by 3 eliminates fractions, which always simplifies equations without changing solutions
2. The multiplication property of equality states that multiplying both sides by the same nonzero number preserves equality (correct answer)
3. The inverse operation property requires multiplying by 3 to cancel the division by 3
4. The distributive property allows multiplying the numerator by 3 while maintaining the equation balance

Explanation: The multiplication property of equality justifies this step: if $a = b$, then $ca = cb$ for any nonzero number $c$. Since 3 ≠ 0, multiplying both sides by 3 preserves the equality. Choice A is incorrect because eliminating fractions doesn't always preserve solutions (e.g., if multiplying by zero). Choice C mentions inverse operations but doesn't explain why the equality is preserved. Choice D misapplies the distributive property.

Question 17

1. The distributive property allows multiplication over subtraction, maintaining equality (correct answer)
2. The associative property allows regrouping terms while preserving the equation
3. The commutative property allows rearranging terms without changing the solution
4. The substitution property allows replacing expressions with equivalent forms

Explanation: The distributive property $a(b - c) = ab - ac$ justifies multiplying 3 by both terms inside the parentheses: $3(x - 4) = 3 \cdot x - 3 \cdot 4 = 3x - 12$. This maintains equality because we're applying the same operation to the same expression. Choice B (associative) deals with grouping, choice C (commutative) deals with order, and choice D (substitution) is too vague for this specific algebraic manipulation.

Question 18

1. Multiplying by 6 eliminates all fractions because 6 is larger than both denominators in the equation
2. Multiplying by 6 is valid because 6 is the least common multiple of 2 and 3, clearing all fractions efficiently
3. Multiplying by 6 applies the multiplication property of equality using the LCM, preserving solutions while eliminating fractions (correct answer)
4. Multiplying by 6 uses the distributive property to separate fractions and create equivalent integer coefficients

Explanation: The complete justification requires two parts: (1) the multiplication property of equality allows multiplying both sides by any nonzero number (here, 6), and (2) choosing 6 (the LCM of 2 and 3) efficiently clears all fractions. This gives $3x + 2x = 30$, or $5x = 30$. Choice A focuses only on size. Choice B mentions LCM but not the equality property. Choice D misapplies the distributive property.

Question 19

1. The distributive property ensures that if a product equals zero, then individual factors can be zero
2. The zero product property states that if $ab = 0$, then $a = 0$ or $b = 0$ (or both) (correct answer)
3. The multiplication property of equality requires that factors be set equal to zero separately
4. The factoring property guarantees that polynomial solutions come from setting factors to zero

Explanation: 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.

Question 20

1. Subtracting $2x$ from both sides maintains equality because equal quantities subtracted from equal quantities remain equal (correct answer)
2. Combining like terms on both sides simplifies the equation while preserving all possible solutions
3. The reflexive property allows moving terms from one side to the other without changing the equation
4. The symmetric property justifies rearranging terms to isolate variables on one side of the equation

Explanation: 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.