19. solve each equation. show all steps.
a) $5x - 3 = 12$
b) $\frac{x}{-4} + 5 = 7$
c) $\frac{18 - p}{4} = -4$
20. solve for the indicated variable. show all work.
a) $v = lwh$ solve for $w$.
b) $y = mx + b$ solve for $x$.
21. a - h) solve each equation. if there is no solution, write no solution. if there are infinite solutions, write identity. show all work.
a) $6v - 4 = -3v + 59$
b) $-t + 7 = 3t + 15$
c) $\frac{2n + 2}{8} = \frac{n}{3}$
d) $9(9 - 4) = 29 - 1$

Question

Accepted Answer

Let's solve each problem step by step:

### Problem 19
#### a) Solve $ 5x - 3 = 12 $
## Step 1: Add 3 to both sides
To isolate the term with $ x $, we add 3 to both sides of the equation.
$ 5x - 3 + 3 = 12 + 3 $
$ 5x = 15 $

## Step 2: Divide by 5
Divide both sides by 5 to solve for $ x $.
$ \frac{5x}{5} = \frac{15}{5} $
$ x = 3 $

#### b) Solve $ \frac{x}{-4} + 5 = 7 $
## Step 1: Subtract 5 from both sides
Subtract 5 from both sides to isolate the fraction.
$ \frac{x}{-4} + 5 - 5 = 7 - 5 $
$ \frac{x}{-4} = 2 $

## Step 2: Multiply by -4
Multiply both sides by -4 to solve for $ x $.
$ x = 2 \times (-4) $
$ x = -8 $

#### c) Solve $ \frac{18 - p}{4} = -4 $
## Step 1: Multiply by 4
Multiply both sides by 4 to eliminate the denominator.
$ 18 - p = -4 \times 4 $
$ 18 - p = -16 $

## Step 2: Subtract 18
Subtract 18 from both sides.
$ -p = -16 - 18 $
$ -p = -34 $

## Step 3: Multiply by -1
Multiply both sides by -1 to solve for $ p $.
$ p = 34 $

### Problem 20
#### a) Solve $ V = lwh $ for $ w $
## Step 1: Divide by $ lh $
To isolate $ w $, divide both sides by $ lh $ (assuming $ l
eq 0 $ and $ h
eq 0 $).
$ w = \frac{V}{lh} $

#### b) Solve $ y = mx + b $ for $ x $
## Step 1: Subtract $ b $
Subtract $ b $ from both sides.
$ y - b = mx $

## Step 2: Divide by $ m $
Divide both sides by $ m $ (assuming $ m
eq 0 $).
$ x = \frac{y - b}{m} $

### Problem 21
#### a) Solve $ 6v - 4 = -3v + 59 $
## Step 1: Add $ 3v $ to both sides
Add $ 3v $ to both sides to get all $ v $ terms on one side.
$ 6v + 3v - 4 = 59 $
$ 9v - 4 = 59 $

## Step 2: Add 4
Add 4 to both sides.
$ 9v = 59 + 4 $
$ 9v = 63 $

## Step 3: Divide by 9
Divide both sides by 9.
$ v = \frac{63}{9} $
$ v = 7 $

#### b) Solve $ -t + 7 = 3t + 15 $
## Step 1: Add $ t $ to both sides
Add $ t $ to both sides to get all $ t $ terms on one side.
$ 7 = 3t + t + 15 $
$ 7 = 4t + 15 $

## Step 2: Subtract 15
Subtract 15 from both sides.
$ 7 - 15 = 4t $
$ -8 = 4t $

## Step 3: Divide by 4
Divide both sides by 4.
$ t = \frac{-8}{4} $
$ t = -2 $

#### c) Solve $ \frac{2n + 2}{8} = \frac{n}{3} $
## Step 1: Cross-multiply
Cross-multiply to eliminate the fractions.
$ 3(2n + 2) = 8n $

## Step 2: Distribute
Distribute the 3 on the left.
$ 6n + 6 = 8n $

## Step 3: Subtract $ 6n $
Subtract $ 6n $ from both sides.
$ 6 = 8n - 6n $
$ 6 = 2n $

## Step 4: Divide by 2
Divide both sides by 2.
$ n = \frac{6}{2} $
$ n = 3 $

#### d) Solve $ 9(9 - 4) = 2g - 1 $
## Step 1: Simplify left side
Simplify the left side (9 - 4 = 5).
$ 9 \times 5 = 2g - 1 $
$ 45 = 2g - 1 $

## Step 2: Add 1
Add 1 to both sides.
$ 45 + 1 = 2g $
$ 46 = 2g $

## Step 3: Divide by 2
Divide both sides by 2.
$ g = \frac{46}{2} $
$ g = 23 $

### Final Answers:
#### Problem 19:
a) $ \boldsymbol{x = 3} $
b) $ \boldsymbol{x = -8} $
c) $ \boldsymbol{p = 34} $

#### Problem 20:
a) $ \boldsymbol{w = \frac{V}{lh}} $
b) $ \boldsymbol{x = \frac{y - b}{m}} $

#### Problem 21:
a) $ \boldsymbol{v = 7} $
b) $ \boldsymbol{t = -2} $
c) $ \boldsymbol{n = 3} $
d) $ \boldsymbol{g = 23} $

QUESTION IMAGE

Question

Explanation:

Problem 19

a) Solve \( 5x - 3 = 12 \)

Step 1: Add 3 to both sides

Step 2: Divide by 5

b) Solve \( \frac{x}{-4} + 5 = 7 \)

Step 1: Subtract 5 from both sides

Step 2: Multiply by -4

c) Solve \( \frac{18 - p}{4} = -4 \)

Step 1: Multiply by 4

Step 2: Subtract 18

Step 3: Multiply by -1

Problem 20

a) Solve \( V = lwh \) for \( w \)

Step 1: Divide by \( lh \)

b) Solve \( y = mx + b \) for \( x \)

Step 1: Subtract \( b \)

Step 2: Divide by \( m \)

Problem 21

a) Solve \( 6v - 4 = -3v + 59 \)

Step 1: Add \( 3v \) to both sides

Step 2: Add 4

Step 3: Divide by 9

b) Solve \( -t + 7 = 3t + 15 \)

Step 1: Add \( t \) to both sides

Step 2: Subtract 15

Step 3: Divide by 4

c) Solve \( \frac{2n + 2}{8} = \frac{n}{3} \)

Step 1: Cross-multiply

Step 2: Distribute

Step 3: Subtract \( 6n \)

Step 4: Divide by 2

d) Solve \( 9(9 - 4) = 2g - 1 \)

Step 1: Simplify left side

Step 2: Add 1

Step 3: Divide by 2

Final Answers:

Problem 19:

Problem 20:

Problem 21:

Answer: