Question

directions: solve each equation for the variable. show your work for credit!

1. $4a - 9 = 7a + 12$
2. $13b = 6b + 14$
3. $3(5c - 4) = 8c + 2$
4. $2d + 18 = 12 - 6(d + 7)$
5. $8(e - 5) = 2(3e - 8)$
6. $f - 8 = 6 + 3f$
7. $g + 4g = 2g - 15$
8. $-3(2h - 4) = 20 - 4h$
9. $-5i - 3 = -43$
10. $j - (5 - 3j) = 7j + 4$
11. $8k - 15 = 4k + 13$
12. $2(5 - 3r) = -4r - 16$

Explanation:

Problem 25: \( 4a - 9 = 7a + 12 \)

Step 1: Subtract \( 4a \) from both sides

To get all \( a \)-terms on one side, we subtract \( 4a \) from both sides.
\( 4a - 9 - 4a = 7a + 12 - 4a \)
Simplifying gives: \( -9 = 3a + 12 \)

Step 2: Subtract 12 from both sides

Now, we isolate the term with \( a \) by subtracting 12 from both sides.
\( -9 - 12 = 3a + 12 - 12 \)
Simplifying gives: \( -21 = 3a \)

Step 3: Divide by 3

To solve for \( a \), we divide both sides by 3.
\( \frac{-21}{3} = \frac{3a}{3} \)
Simplifying gives: \( a = -7 \)

Step 1: Subtract \( 6b \) from both sides

To get all \( b \)-terms on one side, we subtract \( 6b \) from both sides.
\( 13b - 6b = 6b + 14 - 6b \)
Simplifying gives: \( 7b = 14 \)

Step 2: Divide by 7

To solve for \( b \), we divide both sides by 7.
\( \frac{7b}{7} = \frac{14}{7} \)
Simplifying gives: \( b = 2 \)

Step 1: Distribute the 3

First, we distribute the 3 on the left side.
\( 3 \times 5c - 3 \times 4 = 8c + 2 \)
Simplifying gives: \( 15c - 12 = 8c + 2 \)

Step 2: Subtract \( 8c \) from both sides

To get all \( c \)-terms on one side, we subtract \( 8c \) from both sides.
\( 15c - 12 - 8c = 8c + 2 - 8c \)
Simplifying gives: \( 7c - 12 = 2 \)

Step 3: Add 12 to both sides

Now, we isolate the term with \( c \) by adding 12 to both sides.
\( 7c - 12 + 12 = 2 + 12 \)
Simplifying gives: \( 7c = 14 \)

Step 4: Divide by 7

To solve for \( c \), we divide both sides by 7.
\( \frac{7c}{7} = \frac{14}{7} \)
Simplifying gives: \( c = 2 \)

Answer:

\( a = -7 \)

Problem 26: \( 13b = 6b + 14 \)