Question

1. clare was solving an equation, but when she checked her answer, she saw her solution was incorrect. it turns out clares work has three mistakes! 8(2 + 7x)=3x-(2 + 5x) 10 + 56x=3x - 2+5x 10 + 56x=8x - 2 10=64x - 2 12=64x 3/16=x a. find all three mistakes. b. find the solution to the equation.

Explanation:

Step1: Identify first mistake

When distributing 8 in $8(2 + 7x)$, it should be $16+56x$, not $10 + 56x$.

Step2: Identify second mistake

When simplifying $3x-(2 + 5x)$, it should be $3x-2 - 5x=-2x - 2$, not $3x-2 + 5x$.

Step3: Identify third mistake

When going from $10+56x=8x - 2$ to $10 = 64x-2$, a wrong subtraction was done. It should be $10+56x= - 2x-2$, then $10 + 2=-2x-56x$.

Step4: Solve the correct equation

Start with the correct equation $16+56x=-2x - 2$.
Add $2x$ to both sides: $16+56x+2x=-2x - 2+2x$, so $16 + 58x=-2$.
Subtract 16 from both sides: $16-16 + 58x=-2-16$, so $58x=-18$.
Divide both sides by 58: $x=-\frac{18}{58}=-\frac{9}{29}$.

Answer:

a. Mistakes: Incorrect distribution of 8; Incorrect simplification of $3x-(2 + 5x)$; Incorrect subtraction when moving terms.
b. $x =-\frac{9}{29}$