Sure! To solve the system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
x - y = -9 \\
3x + 4y = 8
\end{array}
\right.
\][/tex]
we'll use the substitution or elimination method. Here, I'll demonstrate the substitution method step-by-step:
1. Solve the first equation for one of the variables:
From the first equation [tex]\(x - y = -9\)[/tex], we can solve for [tex]\(x\)[/tex]:
[tex]\[
x = y - 9
\][/tex]
2. Substitute this expression into the second equation:
Now, substitute [tex]\(x = y - 9\)[/tex] into the second equation [tex]\(3x + 4y = 8\)[/tex]:
[tex]\[
3(y - 9) + 4y = 8
\][/tex]
3. Simplify and solve for [tex]\(y\)[/tex]:
Distribute the 3:
[tex]\[
3y - 27 + 4y = 8
\][/tex]
Combine like terms:
[tex]\[
7y - 27 = 8
\][/tex]
Add 27 to both sides:
[tex]\[
7y = 35
\][/tex]
Divide both sides by 7:
[tex]\[
y = 5
\][/tex]
4. Substitute back to find [tex]\(x\)[/tex]:
Substitute [tex]\(y = 5\)[/tex] back into the expression for [tex]\(x\)[/tex]:
[tex]\[
x = y - 9 = 5 - 9 = -4
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (-4, 5)
\][/tex]
This means [tex]\(x = -4\)[/tex] and [tex]\(y = 5\)[/tex].
