Solve the Linear Equation: -2(4+5x) + 3(2-2x) = 8(4-x)

To solve this equation $-2(4 + 5x) + 3(2 - 2x) = 8(4 - x)$, let's work through it step by step:

Step 1: Distribute the constants into the parentheses:

• Distribute $-2$ in $-2(4 + 5x)$:
$-2(4) + -2(5x) = -8 - 10x$
• Distribute $3$ in $3(2 - 2x)$:
$3(2) + 3(-2x) = 6 - 6x$
• Distribute $8$ in $8(4 - x)$:
$8(4) - 8(x) = 32 - 8x$

Step 2: Substitute the distributed expressions back into the equation:

$-8 - 10x + 6 - 6x = 32 - 8x$

Step 3: Combine like terms on each side of the equation:

• On the left side, combine $-8$ and $6$, and $-10x$ and $-6x$:
$-2 - 16x$
• The right side remains:
$32 - 8x$

Step 4: Rewrite the equation:

$-2 - 16x = 32 - 8x$

Step 5: Isolate $x$ by first adding $8x$ to both sides to get rid of $-8x$ on the right:

$-2 - 16x + 8x = 32$ $-2 - 8x = 32$

Step 6: Add $2$ to both sides to isolate the x term:

$-8x = 34$

Step 7: Divide both sides by $-8$ to solve for $x$:

$x = \frac{34}{-8} = -\frac{17}{4}$

Therefore, the solution to the equation is $x = -\frac{17}{4}$.