To solve the system of equations using the addition method, we need to eliminate one of the variables by adding the equations together. Here are the given equations:
[tex]\[
\begin{array}{l}
4x - 5y = -27 \\
3x + 5y = 6
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Write down the system of equations:
[tex]\[
4x - 5y = -27 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
3x + 5y = 6 \quad \text{(Equation 2)}
\][/tex]
2. Add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
(4x - 5y) + (3x + 5y) = -27 + 6
\][/tex]
3. Combine like terms:
[tex]\[
4x + 3x - 5y + 5y = -27 + 6
\][/tex]
[tex]\[
7x = -21
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-21}{7} = -3
\][/tex]
5. Substitute [tex]\( x = -3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
We use Equation 2:
[tex]\[
3x + 5y = 6
\][/tex]
Substituting [tex]\( x = -3 \)[/tex]:
[tex]\[
3(-3) + 5y = 6
\][/tex]
[tex]\[
-9 + 5y = 6
\][/tex]
6. Isolate [tex]\( y \)[/tex]:
[tex]\[
5y = 6 + 9
\][/tex]
[tex]\[
5y = 15
\][/tex]
[tex]\[
y = \frac{15}{5} = 3
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (-3, 3)
\][/tex]
The solution set is [tex]\((-3, 3)\)[/tex].
