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:

$$
LATEXBLOCK0
$$

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).