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📝 Subtraction and Addition Properties of Equality (11 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 11 questions available

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What is Subtraction and Addition Properties of Equality?

Definition:
The Subtraction and Addition Properties of Equality state that if you subtract or add the same quantity from both sides of an equation, the two sides remain equal. These properties are fundamental for isolating a variable to solve for its value. For any real numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a - c = b - c$ and $a + c = b + c$ .

Working:
These properties work by allowing us to perform the inverse operation on one side of the equation while maintaining balance. To solve $x + 5 = 12$ , we use the Subtraction Property to subtract 5 from both sides, giving $x + 5 - 5 = 12 - 5$ , which simplifies to $x = 7$ .

Example:
Solve $y - 3 = 10$ using the Addition Property. Add 3 to both sides: $y - 3 + 3 = 10 + 3$ , so $y = 13$ . Verify by substituting: $13 - 3 = 10$ , which is true.

Reason:
These properties preserve equality, ensuring that any operation performed on both sides keeps the equation balanced. This allows us to undo addition or subtraction and isolate the variable, making it the core method for solving simple linear equations.

Easy

Medium

Hard

📝 All Subtraction and Addition Properties of Equality MCQs

Q1. A student solves the equation $x - 8 = 15$ by adding 8 to both sides, obtaining $x = 23$ . In a second, identical equation, $y - 8 = 15$ , the student subtracts 8 from both sides and gets $y = 7$ . Which statement correctly analyzes this situation?

A.Both methods are valid because addition and subtraction are inverse operations.

B.The first student is correct; the second student violated the Addition Property of Equality. ✅

C.The second student is correct; the first student violated the Subtraction Property of Equality.

D.Both students are incorrect because they should have divided by 8.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The first student correctly applied the Addition Property of Equality by adding 8 to both sides to isolate the variable. The second student incorrectly subtracted 8, which changes the equation to $y - 16 = 7$ , not maintaining equality. This is a classic error of applying the wrong inverse operation.

Q2. A number $n$ satisfies $n + 12 = 5$ . Which real-world scenario best models this equation, and what is the solution?

A.The temperature rose 12 degrees to reach 5°F; original temperature was -7°F. ✅

B.You earned

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' in math mode at position 4: 12 \̲)̲ and now have \…" style="color:#cc0000">12 \) and now have

\</span>5

; you started with

\$17

C.A submarine descends 12 meters to reach 5 meters depth; original depth was 17 meters.

D.You lost 12 pounds and now weigh 5 pounds; original weight was -7 pounds.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The equation $n + 12 = 5$ means something increased by 12 to become 5. The solution is $n = 5 - 12 = -7$ . Only the temperature scenario correctly models an increase (rise of 12 degrees) leading to a final value of 5, with an initial value of -7°F. The other options confuse addition with subtraction or give impossible physical meanings.

Q3. Given the equation $a - 3.7 = 9.2$ , which of the following is the correct first step to solve for $a$ , and what is the reasoning?

A.Add 3.7 to both sides because subtraction is the inverse of addition. ✅

B.Subtract 3.7 from both sides to keep the variable alone.

C.Add 9.2 to both sides to move the constant.

D.Multiply both sides by 3.7 to eliminate the coefficient.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The equation has $a$ minus 3.7. To isolate $a$ , you must perform the inverse operation of subtraction, which is addition. Adding 3.7 to both sides cancels the $-3.7$ on the left, yielding $a = 12.9$ . This demonstrates understanding of inverse operations, not just rote memorization.

Q4. A student writes the steps: $x + 5 = 12 \Rightarrow x + 5 - 5 = 12 - 5 \Rightarrow x = 7$ . Another student writes: $x + 5 = 12 \Rightarrow x = 12 - 5 \Rightarrow x = 7$ . How do these methods compare in terms of the properties of equality?

A.Both are equivalent; the second is a shortcut that still uses the Subtraction Property of Equality. ✅

B.The first is correct; the second is incorrect because it skips writing the property.

C.The first is incorrect because it adds 5 instead of subtracting; the second is correct.

D.Both are incorrect because you must divide by 5 to solve.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The first student explicitly applies the Subtraction Property of Equality by subtracting 5 from both sides. The second student uses the same property implicitly by moving the +5 to the other side as -5, which is the same operation. Both maintain equality and produce the correct solution; the second is a valid algebraic shortcut.

Q5. The graph of $y = x + 4$ is a line. If you solve $x + 4 = 10$ graphically, what is the meaning of the intersection point and how does it relate to the Addition Property?

A.The intersection with

y=10

gives

x=6

; this is because adding 4 to

x

gives 10, so subtracting 4 from 10 solves it. ✅

B.The intersection with the x-axis gives

x=-4

; this is the solution.

C.The intersection with

y=0

gives

x=10

; that is the solution.

D.The line has no intersection because it is parallel to the x-axis.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To solve $x+4=10$ graphically, you find where the line $y=x+4$ crosses the horizontal line $y=10$ . At that point, the x-coordinate satisfies $x+4=10$ . Subtracting 4 from both sides (Addition Property in reverse) gives $x=6$ . This connects the visual representation to the algebraic property.

Q6. A puzzle states: "I think of a number, subtract 15, and get 22. What is my number?" A student solves it as $x - 15 = 22$ and writes $x = 22 - 15 = 7$ . Which of the following correctly analyzes the student's error?

A.The student added instead of subtracted; the correct equation should be

x + 15 = 22

B.The student set up the equation correctly but applied the inverse operation incorrectly; should add 15 to get 37. ✅

C.The student solved correctly because subtracting 15 from 22 gives the original number.

D.The student misread the problem; it should be

15 - x = 22

, giving

x = -7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The correct equation for "subtract 15 from a number to get 22" is $x - 15 = 22$ . To isolate $x$ , you must add 15 to both sides, not subtract. The student incorrectly subtracted 15, yielding 7 instead of the correct 37. This is a common error in translating word problems and applying inverse operations.

Q7. Consider the equation $2x + 7 = 19$ . A student says, "First, I will subtract 7 from both sides because of the Subtraction Property of Equality." Then the student gets $2x = 12$ and divides by 2 to get $x=6$ . Which property is used second, and why is the order important?

A.The second property is the Division Property; order is not important because addition is commutative.

B.The second property is the Addition Property; order matters because you must undo addition before multiplication.

C.The second property is the Division Property; the order is important because you must isolate the variable term before dividing. ✅

D.The second property is the Subtraction Property again; order matters because you must subtract twice.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The first step correctly uses the Subtraction Property to eliminate +7. The second step uses the Division Property to eliminate the coefficient 2. The order is crucial: you must first isolate the variable term (2x) by undoing addition/subtraction, then undo multiplication/division. Reversing the order would make the equation harder to solve correctly.

Q8. Which of the following equations has the same solution as $x - 5 = 12$ after applying the Addition Property of Equality?

x + 5 = 22

x - 5 = 17

x = 17

✅

x + 5 = 12

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Applying the Addition Property of Equality to $x - 5 = 12$ means adding 5 to both sides, resulting in $x = 17$ . This new equation is equivalent to the original and has the same solution. The other options represent different equations that do not directly result from a valid application of the Addition Property to the given equation.

Q9. A student claims that the equations $x + 3 = 7$ and $x = 4$ are not equivalent because one has a +3 and the other doesn't. Which of the following is the best counterargument?

A.They are not equivalent because the variable is different.

B.They are equivalent because subtracting 3 from both sides of the first gives the second, preserving equality. ✅

C.They are equivalent only if you add 3 to both sides of the second.

D.They are not equivalent because the solution sets are different.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student misunderstands equivalence. The Subtraction Property of Equality allows us to subtract the same number from both sides without changing the solution. Starting with $x+3=7$ , subtracting 3 gives $x=4$ . These two equations have the exact same solution set (x=4), so they are equivalent, even though they look different.

Q10. In a math contest, two students solve $2x + 5 = 2x + 8$ . Student A subtracts $2x$ from both sides to get $5=8$ and says no solution. Student B subtracts 5 from both sides first to get $2x = 2x + 3$ , then subtracts $2x$ to get $0=3$ and says no solution. Which student's reasoning is correct, and why?

A.Student A is correct; Student B made an error because you cannot subtract 5 first.

B.Student B is correct; Student A made an error because you must isolate the constant first.

C.Both students are correct; different orders of applying the properties lead to equivalent false statements, both indicating no solution. ✅

D.Neither is correct; the equation has infinitely many solutions because the variable cancels.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both students correctly apply the Subtraction Property of Equality in different orders. Student A subtracts $2x$ first, yielding the false statement $5=8$ . Student B subtracts 5 first, then $2x$ , yielding $0=3$ . In both cases, the variable cancels and a false statement results, meaning the original equation has no solution. This shows that the order of applying properties can vary but the conclusion remains valid.

Q11. A car's fuel efficiency is modeled by $m + 6 = 15$ , where $m$ is miles per gallon. A second car has efficiency modeled by $m - 6 = 3$ . If a student claims both cars have the same efficiency, how would you use the properties of equality to justify or refute this?

A.They are the same because both equations solve to

m = 9

. ✅

B.They are different because the first solves to

m = 21

and the second to

m = -3

C.They are the same because addition and subtraction are inverse operations.

D.They are different because the first solves to

m = 9

and the second to

m = 9

, so they are actually same.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Solving the first equation $m + 6 = 15$ using the Subtraction Property gives $m = 9$ . Solving the second $m - 6 = 3$ using the Addition Property gives $m = 9$ . Both models, although written differently, represent the same fuel efficiency. This shows how different algebraic forms can model the same real quantity and that using the properties correctly reveals their equality.

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