Question

1. 53 = -7 - 10m
2. -8(a - 6) = 104
3. 8(1 + 8r) + 8r = -568

Explanation:

Problem 41: \( 53 = -7 - 10m \)

Step 1: Add 7 to both sides

To isolate the term with \( m \), we add 7 to both sides of the equation. This gives us \( 53 + 7 = -10m \).
Calculating the left side: \( 53 + 7 = 60 \), so the equation becomes \( 60 = -10m \).

Step 2: Divide both sides by -10

To solve for \( m \), we divide both sides of the equation \( 60 = -10m \) by -10.
\( m = \frac{60}{-10} \)

Step 3: Simplify the fraction

Simplifying \( \frac{60}{-10} \) gives us \( m = -6 \).

Step 1: Divide both sides by -8

First, we divide both sides of the equation by -8 to get rid of the coefficient of the parentheses.
\( a - 6 = \frac{104}{-8} \)

Step 2: Simplify the right side

Calculating \( \frac{104}{-8} \) gives us \( -13 \), so the equation becomes \( a - 6 = -13 \).

Step 3: Add 6 to both sides

To solve for \( a \), we add 6 to both sides of the equation \( a - 6 = -13 \).
\( a = -13 + 6 \)

Step 4: Simplify the right side

Simplifying \( -13 + 6 \) gives us \( a = -7 \).

Step 1: Distribute the 8

First, we distribute the 8 in \( 8(1 + 8r) \) to get \( 8 \times 1 + 8 \times 8r = 8 + 64r \). So the equation becomes \( 8 + 64r + 8r = -568 \).

Step 2: Combine like terms

Combining the \( r \)-terms: \( 64r + 8r = 72r \), so the equation is \( 8 + 72r = -568 \).

Step 3: Subtract 8 from both sides

To isolate the term with \( r \), we subtract 8 from both sides: \( 72r = -568 - 8 \).

Step 4: Simplify the right side

Calculating \( -568 - 8 = -576 \), so the equation becomes \( 72r = -576 \).

Step 5: Divide both sides by 72

To solve for \( r \), we divide both sides by 72: \( r = \frac{-576}{72} \).

Step 6: Simplify the fraction

Simplifying \( \frac{-576}{72} \) gives us \( r = -8 \).

Answer:

\( m = -6 \)

Problem 43: \( -8(a - 6) = 104 \)