Certainly! Let's find the solution to the given system of linear equations:
[tex]\[
\begin{cases}
y = 5x + 2 \\
3x = -y + 10
\end{cases}
\][/tex]
To solve this system, we will use the method of substitution. Here's the step-by-step solution:
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
Given:
[tex]\[
y = 5x + 2
\][/tex]
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[
3x = - (5x + 2) + 10
\][/tex]
2. Simplify the equation:
[tex]\[
3x = -5x - 2 + 10
\][/tex]
3. Combine like terms:
[tex]\[
3x = -5x + 8
\][/tex]
4. Add [tex]\( 5x \)[/tex] to both sides of the equation:
[tex]\[
3x + 5x = 8
\][/tex]
[tex]\[
8x = 8
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{8}{8}
\][/tex]
[tex]\[
x = 1
\][/tex]
6. Substitute [tex]\( x \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[
y = 5x + 2
\][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 5(1) + 2
\][/tex]
[tex]\[
y = 5 + 2
\][/tex]
[tex]\[
y = 7
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (1, 7)
\][/tex]
Among the given choices, the correct one is:
[tex]\((1, 7)\)[/tex]
