Question

graph the lines that represent the following system of linear equations.
$2x + 1.5y = -60$
$2x + y = -50$
then, mark the point on the graph that represents the solution to the system.

Explanation:

Step1: Solve the system of equations

We have the system:
\[

$$\begin{cases} 2x + 1.5y = -60 \\ 2x + y = -50 \end{cases}$$

\]
Subtract the second equation from the first equation to eliminate \(x\):
\[
(2x + 1.5y) - (2x + y) = -60 - (-50)
\]
\[
2x + 1.5y - 2x - y = -60 + 50
\]
\[
0.5y = -10
\]
Multiply both sides by 2:
\[
y = -20
\]
Substitute \(y = -20\) into the second equation \(2x + y = -50\):
\[
2x + (-20) = -50
\]
\[
2x - 20 = -50
\]
Add 20 to both sides:
\[
2x = -50 + 20
\]
\[
2x = -30
\]
Divide both sides by 2:
\[
x = -15
\]
So the solution to the system is the point \((-15, -20)\).

Step2: Graph the first line \(2x + 1.5y = -60\)

Rewrite it in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
\[
1.5y=-2x - 60
\]
\[
y=\frac{-2}{1.5}x-\frac{60}{1.5}
\]
\[
y =-\frac{4}{3}x - 40
\]
To find two points on the line, when \(x = 0\), \(y=-40\). When \(y = 0\), \(0=-\frac{4}{3}x-40\), \(\frac{4}{3}x=-40\), \(x=-30\). So two points on the first line are \((0, - 40)\) and \((-30,0)\).

Step3: Graph the second line \(2x + y=-50\)

Rewrite it in slope - intercept form:
\[
y=-2x - 50
\]
When \(x = 0\), \(y=-50\). When \(y = 0\), \(0=-2x-50\), \(2x=-50\), \(x=-25\). So two points on the second line are \((0,-50)\) and \((-25,0)\).

Step4: Mark the solution point

The solution to the system of equations is the point of intersection of the two lines. From our calculation, the point is \((-15,-20)\). We mark this point on the graph.

Answer:

The solution to the system of linear equations is the point \((-15, - 20)\). To graph the lines:

• For \(2x + 1.5y=-60\) (or \(y =-\frac{4}{3}x - 40\)), plot the points \((0,-40)\) and \((-30,0)\) and draw a line through them.
• For \(2x + y=-50\) (or \(y=-2x - 50\)), plot the points \((0,-50)\) and \((-25,0)\) and draw a line through them.
• Then mark the point \((-15, - 20)\) as the intersection point (the solution).