Certainly! Let's solve the system of equations using the substitution method step-by-step. The system of equations provided is:
[tex]\[
\begin{cases}
3x + 2y = -9 & \quad (1) \\
x = y + 7 & \quad (2)
\end{cases}
\][/tex]
### Step 1: Solve one of the equations for one variable
Equation (2) is already solved for [tex]\(x\)[/tex]:
[tex]\[
x = y + 7
\][/tex]
### Step 2: Substitute this expression into the other equation
We will substitute [tex]\(x = y + 7\)[/tex] into equation (1):
[tex]\[
3(y + 7) + 2y = -9
\][/tex]
### Step 3: Simplify the resulting equation
First, distribute the 3 on the left side:
[tex]\[
3y + 21 + 2y = -9
\][/tex]
Combine like terms:
[tex]\[
5y + 21 = -9
\][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
Subtract 21 from both sides to isolate the [tex]\(y\)[/tex] term:
[tex]\[
5y = -9 - 21
\][/tex]
[tex]\[
5y = -30
\][/tex]
Divide both sides by 5:
[tex]\[
y = -6
\][/tex]
### Step 5: Substitute [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex]
We found [tex]\(y = -6\)[/tex]. Substitute this value into the expression [tex]\(x = y + 7\)[/tex]:
[tex]\[
x = -6 + 7
\][/tex]
[tex]\[
x = 1
\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[
(x, y) = (1, -6)
\][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[
x = 1, \quad y = -6
\][/tex]
