To solve the system of linear equations using the addition method, follow these steps:
Given the system:
[tex]\[
\left\{\begin{array}{l}
9x - 9y = -90 \\
2x + 9y = 57
\end{array}\right.
\][/tex]
1. Add the Equations:
First, we attempt to eliminate one variable by adding the two equations. Notably, the coefficients of [tex]\( y \)[/tex] in the two equations are already opposites ([tex]\(-9\)[/tex] and [tex]\(9\)[/tex]). Therefore, adding these two equations will eliminate [tex]\( y \)[/tex].
[tex]\[
\begin{array}{l}
(9x - 9y) + (2x + 9y) = -90 + 57
\end{array}
\][/tex]
2. Simplify the Result:
Simplify the left-hand side and the right-hand side of the resulting equation:
[tex]\[
9x - 9y + 2x + 9y = -90 + 57
\][/tex]
Notice that [tex]\(-9y\)[/tex] and [tex]\(9y\)[/tex] cancel each other out.
[tex]\[
9x + 2x = -90 + 57
\][/tex]
[tex]\[
11x = -33
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Now, solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 11:
[tex]\[
x = \frac{-33}{11}
\][/tex]
[tex]\[
x = -3
\][/tex]
So, we have [tex]\( x = -3 \)[/tex].
4. Substitute [tex]\( x \)[/tex] Back into One of the Original Equations:
Substitute [tex]\( x = -3 \)[/tex] into either of the original equations to solve for [tex]\( y \)[/tex].
Using the second equation [tex]\( 2x + 9y = 57 \)[/tex]:
[tex]\[
2(-3) + 9y = 57
\][/tex]
Simplify the left-hand side:
[tex]\[
-6 + 9y = 57
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
Isolate [tex]\( y \)[/tex] by adding 6 to both sides of the equation:
[tex]\[
9y = 57 + 6
\][/tex]
[tex]\[
9y = 63
\][/tex]
Divide both sides by 9:
[tex]\[
y = \frac{63}{9}
\][/tex]
[tex]\[
y = 7
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (-3, 7)
\][/tex]
One solution: [tex]\((-3, 7)\)[/tex]
