To solve the equations, we'll start with each one individually.
1. Solve the equation \( 5(a - 2) = 2 + 3a \)
First, distribute the left side:
\[ 5a - 10 = 2 + 3a \]
Next, subtract \( 3a \) from both sides:
\[ 5a - 3a - 10 = 2 \]
This simplifies to:
\[ 2a - 10 = 2 \]
Now, add 10 to both sides:
\[ 2a = 12 \]
Now divide both sides by 2:
\[ a = 6 \]
2. Solve the equation \( -8(-5a + 7) = 7(7r + 1) \)
Distribute on both sides:
\[ 40a - 56 = 49r + 7 \]
Now, let's isolate terms involving \( a \) and \( r \). Add 56 to both sides:
\[ 40a = 49r + 63 \]
This can also be expressed as:
\[ 40a - 49r = 63 \]
This is a linear equation with two variables, so we cannot solve for specific values of \( a \) or \( r \) without additional information.
3. Solve the equation \( x - 29 = -3(3 + x) \)
First, distribute on the right:
\[ x - 29 = -9 - 3x \]
Now, add \( 3x \) to both sides:
\[ x + 3x - 29 = -9 \]
This simplifies to:
\[ 4x - 29 = -9 \]
Next, add 29 to both sides:
\[ 4x = 20 \]
Now, divide both sides by 4:
\[ x = 5 \]
Summarizing the solutions:
- \( a = 6 \)
- \( 40a - 49r = 63 \) (relationship between \( a \) and \( r \))
- \( x = 5 \)