Let's carefully examine the student's work and identify the mistake. We are given the following system of linear equations:
[tex]\[
\begin{array}{l}
3x + y = 9 \\
-2x + y = 4
\end{array}
\][/tex]
The student's work involves solving the second equation for [tex]\( y \)[/tex]:
1. The second equation is [tex]\(-2x + y = 4\)[/tex]. Solving for [tex]\( y \)[/tex], we get:
[tex]\[
y = 2x + 4
\][/tex]
2. The student then substitutes [tex]\( y = 2x + 4 \)[/tex] into the first equation:
[tex]\[
3x + (2x + 4) = 9
\][/tex]
3. Simplifying the equation:
[tex]\[
3x + 2x + 4 = 9
\][/tex]
[tex]\[
5x + 4 = 9
\][/tex]
4. Subtracting 4 from both sides:
[tex]\[
5x = 5
\][/tex]
5. Dividing by 5:
[tex]\[
x = 1
\][/tex]
6. Substitute [tex]\( x = 1 \)[/tex] back into the expression [tex]\( y = 2x + 4 \)[/tex]:
[tex]\[
y = 2(1) + 4
\][/tex]
[tex]\[
y = 2 + 4
\][/tex]
[tex]\[
y = 6
\][/tex]
Therefore, the solution to the system of equations is [tex]\( x = 1 \)[/tex] and [tex]\( y = 6 \)[/tex].
The mistake the student made was stopping at the equation [tex]\( 4 = 4 \)[/tex], which is always true and led them to incorrectly conclude that there are infinitely many solutions. This step should have been used to determine the specific value of [tex]\( x \)[/tex], and subsequently, [tex]\( y \)[/tex], to find the unique solution to the system. The correct solution is [tex]\( x = 1 \)[/tex] and [tex]\( y = 6 \)[/tex], indicating that there is exactly one solution, not infinitely many.
