To solve the system of equations:
[tex]\[
\begin{array}{l}
y = -\frac{1}{2}x - 7 \\
-2y = x + 14
\end{array}
\][/tex]
we'll follow these steps:
### Step 1: Graph the Equations
Equation 1: [tex]\( y = -\frac{1}{2}x - 7 \)[/tex]
This is a linear equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]) where [tex]\( m = -\frac{1}{2} \)[/tex] is the slope and [tex]\( b = -7 \)[/tex] is the y-intercept.
Plotting Points:
1. Start at the y-intercept [tex]\((0, -7)\)[/tex].
2. Use the slope [tex]\(-\frac{1}{2}\)[/tex] to find another point. Move down 1 unit and right 2 units from [tex]\((0, -7)\)[/tex] to [tex]\((2, -8)\)[/tex].
Draw a line through these points.
Equation 2: [tex]\( -2y = x + 14 \)[/tex]
We need to rewrite this in slope-intercept form ([tex]\( y = mx + b \)[/tex]).
[tex]\[
-2y = x + 14 \implies y = -\frac{1}{2}x - 7
\][/tex]
So, this simplifies to:
[tex]\[
y = -\frac{1}{2}x - 7
\][/tex]
### Step 2: Analyzing the System
When we compare both equations, we see:
[tex]\[
y = -\frac{1}{2}x - 7
\][/tex]
[tex]\[
y = -\frac{1}{2}x - 7
\][/tex]
Both equations are identical, meaning they represent the same line. Therefore, every point on the line satisfies both equations.
### Conclusion
Since both equations represent the same line, there are infinitely many solutions—every point on the line [tex]\( y = -\frac{1}{2}x - 7 \)[/tex] is a solution to the system.
The correct answer is:
[tex]\[
\text{solutions: all numbers on the line}
\][/tex]
