Select the correct answer.
Paula used these steps to solve an equation:
Step 1: $-4(x+8)-2 x=25$
Step 2: $-4 x-32-2 x=25$
Step 3: $-6 x-32=25$
Step 4: $-6 x=57$
Step 5: $x=-9 \frac{1}{2}$
Between which two steps did Paula use the division property of equality?
A. steps 1 and 2
B. steps 2 and 3
C. steps 3 and 4
D. steps 4 and 5
Answer (1)
Step 1 uses the distributive property.
Step 2 combines like terms.
Step 3 adds 32 to both sides.
Step 4 divides both sides by -6, using the division property of equality, to get x = − 9 2 1 .
Explanation
Understanding the Problem We are given an equation that Paula solved in 5 steps and we need to determine between which two steps she used the division property of equality. The division property of equality states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced.
Analyzing Each Step Let's examine each step to see where the division property was applied.
Step 1 to Step 2: − 4 ( x + 8 ) − 2 x = 25 becomes − 4 x − 32 − 2 x = 25 . This step uses the distributive property: − 4 ( x + 8 ) = − 4 x − 32 .
Step 2 to Step 3: − 4 x − 32 − 2 x = 25 becomes − 6 x − 32 = 25 . This step combines like terms: − 4 x − 2 x = − 6 x .
Step 3 to Step 4: − 6 x − 32 = 25 becomes − 6 x = 57 . This step adds 32 to both sides of the equation: − 6 x − 32 + 32 = 25 + 32 , which simplifies to − 6 x = 57 .
Step 4 to Step 5: − 6 x = 57 becomes x = − 9 2 1 . This step divides both sides of the equation by -6: − 6 − 6 x = − 6 57 , which simplifies to x = − 6 57 = − 2 19 = − 9 2 1 .
Identifying the Correct Step The division property of equality was used between Step 4 and Step 5.
Examples
The division property of equality is a fundamental concept in algebra. Imagine you're splitting a bill equally among friends. If the total bill is 60 an d t h ere a re 5 f r i e n d s , yo u d i v i d e t h e t o t a l cos t b y t h e n u mb ero ff r i e n d s t o f in d e a c h p erso n ′ ss ha re : \frac{60}{5} = 12$. Each friend owes $12. This is a real-world application of the division property of equality, ensuring fairness in splitting costs or resources.
