Question

select the correct answer.
what is the justification for step 2 in the solution process?
10x - 25 - 3x = 4x - 1
step 1: 7x - 25 = 4x - 1
step 2: 7x = 4x + 24
a. the multiplication property of equality
b. the division property of equality
c. the addition property of equality
d. the subtraction property of equality

Explanation:

Step 1: Analyze Step 1 and Step 2 Equations

Step 1: \(7x - 25 = 4x - 1\)
Step 2: \(7x = 4x + 24\)
To get from Step 1 to Step 2, we add 25 to both sides of the equation (to eliminate -25 on the left) and add 1 to both sides (to eliminate -1 on the right, which is equivalent to adding 24 overall? Wait, no—let's check the arithmetic. Wait, actually, if we take Step 1: \(7x - 25 = 4x - 1\), and we add 25 to both sides: \(7x - 25 + 25 = 4x - 1 + 25\), which simplifies to \(7x = 4x + 24\). So we added 25 to both sides. The addition property of equality states that if \(a = b\), then \(a + c = b + c\) for any real number \(c\). Here, \(c = 25\) (added to both sides). Wait, but the result is \(7x = 4x + 24\), which is consistent with adding 25 to both sides (since \(-1 + 25 = 24\)). So the operation is adding 25 to both sides, which is the addition property of equality. Wait, but let's check the options. Option C is "the addition property of equality". Wait, but let's re-examine. Wait, Step 1: \(7x -25 = 4x -1\). Step 2: \(7x = 4x +24\). So we added 25 to both sides (to cancel -25 on left) and added 1 to both sides? No, actually, adding 25 to both sides: left side becomes \(7x -25 +25 = 7x\), right side becomes \(4x -1 +25 = 4x +24\). So that's adding 25 to both sides, which is the addition property of equality. So the justification is the addition property of equality, which is option C. Wait, but let's check the options again. Option C is "the addition property of equality". So that's the correct one.

Step 2: Match with Equality Properties

The addition property of equality allows us to add the same number to both sides of an equation, preserving equality. Here, we added 25 to both sides (and -1 +25 =24, so the right side becomes 4x +24). So the justification for Step 2 is the addition property of equality.

Answer:

C. the addition property of equality