Sure, let's solve this system of linear equations step by step.
We are given the system of equations:
[tex]\[
\begin{cases}
-7x - 2y = 14 \\
6x + 6y = 18
\end{cases}
\][/tex]
### Step 1: Simplify the Equations if Possible
First, we’ll simplify the second equation:
[tex]\[
6x + 6y = 18
\][/tex]
Divide every term by 6:
[tex]\[
x + y = 3 \quad \text{(Equation 2)}
\][/tex]
### Step 2: Use Substitution or Elimination
We can use the substitution method here. Express [tex]\(y\)[/tex] from Equation 2:
[tex]\[
y = 3 - x \quad \text{(Equation 3)}
\][/tex]
### Step 3: Substitute into the First Equation
Now, substitute Equation 3 into the first equation:
[tex]\[
-7x - 2(3 - x) = 14
\][/tex]
Simplify this equation:
[tex]\[
-7x - 6 + 2x = 14
\][/tex]
Combine like terms:
[tex]\[
-5x - 6 = 14
\][/tex]
Add 6 to both sides:
[tex]\[
-5x = 20
\][/tex]
Divide both sides by -5:
[tex]\[
x = -4
\][/tex]
### Step 4: Substitute [tex]\(x\)[/tex] Back into Equation 3
Now, substitute [tex]\(x = -4\)[/tex] back into Equation 3:
[tex]\[
y = 3 - (-4)
\][/tex]
Simplify:
[tex]\[
y = 3 + 4 \\
y = 7
\][/tex]
### Step 5: Write the Solution
Therefore, the solution to the system of equations is:
[tex]\[
x = -4
\][/tex]
[tex]\[
y = 7
\][/tex]
We can now fill in the blanks:
[tex]\[
x = -4
\][/tex]
[tex]\[
y = 7
\][/tex]
This means the solution to the system of equations is [tex]\((-4, 7)\)[/tex].
