solving equations ii
1. solve each equation. be sure to show all necessary steps.
$39 - 12w = 7 - 16w$
$63 - x = 2x + 3$
$12r - 18 = 13r + 18$
$-15v - 40 = 23 - 8v$
2. solve each equation. then perform a ls/rs check.
$-x - 1 = x - 21$
$3n + 46 = 1 + 8n$

Question

Accepted Answer

### 1. Solve $ 39 - 12w = 7 - 16w $
# Explanation:
## Step1: Add $ 16w $ to both sides
To get the variable terms on one side, we add $ 16w $ to both sides of the equation. This gives us $ 39 - 12w + 16w = 7 - 16w + 16w $, which simplifies to $ 39 + 4w = 7 $.
## Step2: Subtract 39 from both sides
Now, we want to isolate the term with the variable. Subtract 39 from both sides: $ 39 + 4w - 39 = 7 - 39 $, which simplifies to $ 4w = -32 $.
## Step3: Divide by 4
To solve for $ w $, divide both sides by 4: $ \frac{4w}{4} = \frac{-32}{4} $, so $ w = -8 $.
# Answer:
$ w = -8 $

### 2. Solve $ 63 - x = 2x + 3 $
# Explanation:
## Step1: Add $ x $ to both sides
Add $ x $ to both sides to get the $ x $ terms on one side: $ 63 - x + x = 2x + 3 + x $, which simplifies to $ 63 = 3x + 3 $.
## Step2: Subtract 3 from both sides
Subtract 3 from both sides: $ 63 - 3 = 3x + 3 - 3 $, so $ 60 = 3x $.
## Step3: Divide by 3
Divide both sides by 3: $ \frac{60}{3} = \frac{3x}{3} $, so $ x = 20 $.
# Answer:
$ x = 20 $

### 3. Solve $ 12r - 18 = 13r + 18 $
# Explanation:
## Step1: Subtract $ 12r $ from both sides
Subtract $ 12r $ from both sides: $ 12r - 18 - 12r = 13r + 18 - 12r $, which simplifies to $ -18 = r + 18 $.
## Step2: Subtract 18 from both sides
Subtract 18 from both sides: $ -18 - 18 = r + 18 - 18 $, so $ r = -36 $.
# Answer:
$ r = -36 $

### 4. Solve $ -15v - 40 = 23 - 8v $
# Explanation:
## Step1: Add $ 15v $ to both sides
Add $ 15v $ to both sides: $ -15v - 40 + 15v = 23 - 8v + 15v $, which simplifies to $ -40 = 23 + 7v $.
## Step2: Subtract 23 from both sides
Subtract 23 from both sides: $ -40 - 23 = 23 + 7v - 23 $, so $ -63 = 7v $.
## Step3: Divide by 7
Divide both sides by 7: $ \frac{-63}{7} = \frac{7v}{7} $, so $ v = -9 $.
# Answer:
$ v = -9 $

### 5. Solve $ -x - 1 = x - 21 $ and check
# Explanation:
## Step1: Add $ x $ to both sides
Add $ x $ to both sides: $ -x - 1 + x = x - 21 + x $, which simplifies to $ -1 = 2x - 21 $.
## Step2: Add 21 to both sides
Add 21 to both sides: $ -1 + 21 = 2x - 21 + 21 $, so $ 20 = 2x $.
## Step3: Divide by 2
Divide both sides by 2: $ \frac{20}{2} = \frac{2x}{2} $, so $ x = 10 $.
## Step4: Check LS and RS
Left Side (LS): $ -x - 1 = -10 - 1 = -11 $
Right Side (RS): $ x - 21 = 10 - 21 = -11 $
Since LS = RS, the solution is correct.
# Answer:
$ x = 10 $

### 6. Solve $ 3n + 46 = 1 + 8n $ and check
# Explanation:
## Step1: Subtract $ 3n $ from both sides
Subtract $ 3n $ from both sides: $ 3n + 46 - 3n = 1 + 8n - 3n $, which simplifies to $ 46 = 1 + 5n $.
## Step2: Subtract 1 from both sides
Subtract 1 from both sides: $ 46 - 1 = 1 + 5n - 1 $, so $ 45 = 5n $.
## Step3: Divide by 5
Divide both sides by 5: $ \frac{45}{5} = \frac{5n}{5} $, so $ n = 9 $.
## Step4: Check LS and RS
Left Side (LS): $ 3n + 46 = 3(9) + 46 = 27 + 46 = 73 $
Right Side (RS): $ 1 + 8n = 1 + 8(9) = 1 + 72 = 73 $
Since LS = RS, the solution is correct.
# Answer:
$ n = 9 $

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QUESTION IMAGE

Question

Explanation:

1. Solve \( 39 - 12w = 7 - 16w \)

Step1: Add \( 16w \) to both sides

Step2: Subtract 39 from both sides

Step3: Divide by 4

Step1: Add \( x \) to both sides

Step2: Subtract 3 from both sides

Step3: Divide by 3

Step1: Subtract \( 12r \) from both sides

Step2: Subtract 18 from both sides

Answer:

2. Solve \( 63 - x = 2x + 3 \)