Question

topic 1: multi-step equations

1. $8 - 3(k + 2) = 2 - 3k$
2. $-7(a - 3) = 11 - 7a$
3. $4(4y - 3) - (y - 5) = -52$

topic 2: proportions

1. $\frac{6}{a - 6} = \frac{3}{a - 3}$
2. $\frac{6}{2n - 5} = \frac{2}{n - 7}$

Explanation:

Problem 2: \( 8 - 3(k + 2) = 2 - 3k \)

Step 1: Distribute the -3

\( 8 - 3k - 6 = 2 - 3k \)

Step 2: Simplify left side

\( 2 - 3k = 2 - 3k \)

Step 3: Analyze the equation

Subtract \( 2 - 3k \) from both sides: \( 0 = 0 \)
This is always true, meaning the equation has infinitely many solutions. But the handwritten note says "no solution" which is incorrect. The correct conclusion is all real numbers are solutions.

Step 1: Distribute the -7

\( -7a + 21 = 11 - 7a \)

Step 2: Add \( 7a \) to both sides

\( 21 = 11 \)

Step 3: Analyze the result

\( 21 = 11 \) is false, so the equation has no solution.

Step 1: Distribute the 4 and the negative sign

\( 16y - 12 - y + 5 = -52 \)

Step 2: Combine like terms

\( 15y - 7 = -52 \)

Step 3: Add 7 to both sides

\( 15y = -45 \)

Step 4: Divide by 15

\( y = -3 \)

Answer:

All real numbers (or infinitely many solutions)

Problem 4: \( -7(a - 3) = 11 - 7a \)