To solve the system of equations:
[tex]\[
\left\{\begin{array}{l}
y = -\frac{1}{2} x - 6 \\
2y - 3x = -8
\end{array}\right.
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here is a step-by-step process to solve this system:
### Step 1: Express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex] from the first equation
The first equation is already solved for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{1}{2}x - 6
\][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] into the second equation
Now, we substitute [tex]\(y\)[/tex] from the first equation into the second equation:
[tex]\[
2 \left(-\frac{1}{2} x - 6\right) - 3x = -8
\][/tex]
### Step 3: Simplify the equation
We first distribute [tex]\(2\)[/tex] to both terms inside the parenthesis:
[tex]\[
2 \left(-\frac{1}{2} x\right) + 2(-6) - 3x = -8
\][/tex]
[tex]\[
- x - 12 - 3x = -8
\][/tex]
### Step 4: Combine like terms
Combine all the [tex]\(x\)[/tex] terms:
[tex]\[
-4x - 12 = -8
\][/tex]
### Step 5: Isolate [tex]\(x\)[/tex]
First, add 12 to both sides of the equation:
[tex]\[
-4x - 12 + 12 = -8 + 12
\][/tex]
[tex]\[
-4x = 4
\][/tex]
Next, divide both sides by [tex]\(-4\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4}{-4}
\][/tex]
[tex]\[
x = -1
\][/tex]
### Step 6: Substitute [tex]\(x\)[/tex] back into the first equation to find [tex]\(y\)[/tex]
We substitute [tex]\(x = -1\)[/tex] back into the first equation:
[tex]\[
y = -\frac{1}{2}(-1) - 6
\][/tex]
[tex]\[
y = \frac{1}{2} - 6
\][/tex]
[tex]\[
y = \frac{1}{2} - \frac{12}{2}
\][/tex]
[tex]\[
y = -\frac{11}{2}
\][/tex]
[tex]\[
y = -5.5
\][/tex]
### Conclusion:
The solution to the system of equations is:
[tex]\[
x = -1 \quad \text{and} \quad y = -5.5
\][/tex]
Thus, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\(x = -1\)[/tex] and [tex]\(y = -5.5\)[/tex].
