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!

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!

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!

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.'

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!

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!

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!

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!

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!

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!