To solve the system of equations given by:
[tex]\[
y = \frac{1}{2}x + 4
\][/tex]
and
[tex]\[
y = -x - 2
\][/tex]
we will proceed step by step to find the solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### Step 1: Set the Equations Equal
Since both equations equal [tex]\(y\)[/tex], we can set them equal to each other:
[tex]\[
\frac{1}{2}x + 4 = -x - 2
\][/tex]
### Step 2: Solve for [tex]\(x\)[/tex]
To eliminate the fractions and simplify the equation, we start by isolating [tex]\(x\)[/tex]. First, we add [tex]\(x\)[/tex] to both sides to get rid of the [tex]\(x\)[/tex] on the right side:
[tex]\[
\frac{1}{2}x + x + 4 = -2
\][/tex]
Combine like terms on the left side:
[tex]\[
\frac{3}{2}x + 4 = -2
\][/tex]
Next, subtract 4 from both sides to isolate the term with [tex]\(x\)[/tex] on the left:
[tex]\[
\frac{3}{2}x = -2 - 4
\][/tex]
Simplify the right side:
[tex]\[
\frac{3}{2}x = -6
\][/tex]
To solve for [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{3}{2}\)[/tex], which is [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
x = -6 \cdot \frac{2}{3}
\][/tex]
Simplify the multiplication:
[tex]\[
x = -4
\][/tex]
### Step 3: Solve for [tex]\(y\)[/tex]
Now that we have [tex]\(x = -4\)[/tex], we can substitute this value back into either of the original equations to solve for [tex]\(y\)[/tex]. We'll use the first equation:
[tex]\[
y = \frac{1}{2}x + 4
\][/tex]
Substitute [tex]\(x = -4\)[/tex]:
[tex]\[
y = \frac{1}{2}(-4) + 4
\][/tex]
Simplify the expression:
[tex]\[
y = -2 + 4
\][/tex]
[tex]\[
y = 2
\][/tex]
### Solution
So, the solution to the system of equations is:
[tex]\[
x = -4 \quad \text{and} \quad y = 2
\][/tex]
### Verification
To verify, we can substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 2\)[/tex] into both original equations to ensure they hold true.
1. For [tex]\( y = \frac{1}{2}x + 4 \)[/tex]:
[tex]\[
2 = \frac{1}{2}(-4) + 4 \implies 2 = -2 + 4 \implies 2 = 2
\][/tex]
2. For [tex]\( y = -x - 2 \)[/tex]:
[tex]\[
2 = -(-4) - 2 \implies 2 = 4 - 2 \implies 2 = 2
\][/tex]
Both equations are satisfied, confirming the solution is correct.
Thus, the solution to the system of equations is [tex]\( \boxed{ x = -4, y = 2 } \)[/tex].
