Let's carefully examine each step of solving the system of linear equations to determine where Michael made his first error.
We start with the system of equations:
[tex]\[
\begin{aligned}
3x - 3y &= 6 \quad \text{(Equation 1)}\\
4x - 7y &= 2 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
Step 1: Multiply the equations to eliminate [tex]\( x \)[/tex].
Multiply Equation 1 by 4:
[tex]\[
4(3x - 3y) = 4(6) \implies 12x - 12y = 24
\][/tex]
Multiply Equation 2 by -3:
[tex]\[
-3(4x - 7y) = -3(2) \implies -12x + 21y = -6
\][/tex]
This step is correct. Let's move on to Step 2.
Step 2: Add the two equations to eliminate [tex]\( x \)[/tex].
[tex]\[
\begin{aligned}
12x - 12y &= 24 \quad \text{(Equation 3)}\\
-12x + 21y &= -6 \quad \text{(Equation 4)}
\end{aligned}
\][/tex]
Adding Equation 3 and Equation 4:
[tex]\[
(12x - 12y) + (-12x + 21y) = 24 + (-6) \implies 9y = 18
\][/tex]
This step should yield:
[tex]\[
9y = 18 \implies y = 2
\][/tex]
Michael, however, showed:
[tex]\[
-33y = 30
\][/tex]
This means Michael made an error in Step 2.
Given that:
- The incorrect intermediate value for [tex]\( y \)[/tex] was [tex]\( -\frac{10}{11} \)[/tex],
- The correct value for [tex]\( y \)[/tex] from the corrected steps is [tex]\( 2 \)[/tex],
- The correct further substitution for [tex]\( x \)[/tex],
The error occurred in Step 2 when combining Equations 3 and 4. Therefore, Michael's first error happened in Step 2.
