To determine the graph that represents the given system of equations, we can follow a step-by-step process to understand the graphical representation. The system of equations is:
1. [tex]\(-2x - y = 10\)[/tex]
2. [tex]\(x - 2y = 5\)[/tex]
### Step 1: Solve each equation for [tex]\(y\)[/tex]
We'll start by solving both equations for [tex]\(y\)[/tex] to make it easier to plot.
#### Equation 1: [tex]\(-2x - y = 10\)[/tex]
Add [tex]\(2x\)[/tex] to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
- y = 10 + 2x
\][/tex]
Multiply both sides by [tex]\(-1\)[/tex]:
[tex]\[
y = -2x - 10
\][/tex]
#### Equation 2: [tex]\(x - 2y = 5\)[/tex]
Subtract [tex]\(x\)[/tex] from both sides to isolate terms with [tex]\(y\)[/tex]:
[tex]\[
-2y = 5 - x
\][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = \frac{x - 5}{2}
\][/tex]
### Step 2: Identify the equations in slope-intercept form
Now, we have the equations in slope-intercept form ([tex]\(y = mx + b\)[/tex]):
1. [tex]\(y = -2x - 10\)[/tex] (Equation 1)
2. [tex]\(y = \frac{1}{2}x - \frac{5}{2}\)[/tex] (Equation 2)
### Step 3: Plot the equations on the coordinate plane
To plot the equations, we need to find the y-intercept ([tex]\(b\)[/tex]) and the slope ([tex]\(m\)[/tex]) for each line:
#### Equation 1: [tex]\(y = -2x - 10\)[/tex]
- Slope ([tex]\(m\)[/tex]): [tex]\(-2\)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): [tex]\(-10\)[/tex]
This line decreases steeply and crosses the y-axis at [tex]\(-10\)[/tex].
#### Equation 2: [tex]\(y = \frac{1}{2}x - \frac{5}{2}\)[/tex]
- Slope ([tex]\(m\)[/tex]): [tex]\(\frac{1}{2}\)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): [tex]\(-\frac{5}{2} = -2.5\)[/tex]
This line rises gradually and crosses the y-axis at [tex]\(-2.5\)[/tex].
### Step 4: Determine the point of intersection
The solution to the system of equations is the point where the two lines intersect. To find the intersection, we can solve the system algebraically by setting the two expressions for [tex]\(y\)[/tex] equal to each other:
[tex]\[
-2x - 10 = \frac{x - 5}{2}
\][/tex]
Multiply every term by 2 to eliminate the fraction:
[tex]\[
-4x - 20 = x - 5
\][/tex]
Combine like terms:
[tex]\[
-4x - x = -5 + 20
\][/tex]
[tex]\[
-5x = 15
\][/tex]
Divide by -5:
[tex]\[
x = -3
\][/tex]
Now substitute [tex]\(x = -3\)[/tex] back into Equation 1 to find [tex]\(y\)[/tex]:
[tex]\[
y = -2(-3) - 10
\][/tex]
[tex]\[
y = 6 - 10
\][/tex]
[tex]\[
y = -4
\][/tex]
### Step 5: Conclusion
The point of intersection is [tex]\((-3, -4)\)[/tex].
So, the correct graph must have two lines that intersect at [tex]\((-3, -4)\)[/tex]. By plotting these lines using their slopes and intercepts, we visualize the system in the coordinate plane. Ensure to look for the point of intersection [tex]\((-3, -4)\)[/tex] when choosing the correct graph.
