To solve the system of linear equations, we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. The given system of equations is:
[tex]\[
\begin{cases}
y = -2x + 7 \\
y = 5x - 7
\end{cases}
\][/tex]
Here's the step-by-step solution:
1. Set the equations equal to each other since both are equal to [tex]\(y\)[/tex]:
[tex]\[
-2x + 7 = 5x - 7
\][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[
7 + 7 = 5x + 2x
\][/tex]
[tex]\[
14 = 7x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{14}{7}
\][/tex]
[tex]\[
x = 2
\][/tex]
Now that we have determined [tex]\(x = 2\)[/tex], we need to find the corresponding value of [tex]\(y\)[/tex].
4. Substitute [tex]\(x = 2\)[/tex] back into one of the original equations. We'll use the first equation:
[tex]\[
y = -2(2) + 7
\][/tex]
[tex]\[
y = -4 + 7
\][/tex]
[tex]\[
y = 3
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\begin{cases}
x = 2 \\
y = 3
\end{cases}
\][/tex]
So, the solution is [tex]\( \boxed{(2, 3)} \)[/tex].
