Question

question
solve the following system of equations graphically on the set of axes below.
$y = 3x - 8$
$y = -2x - 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze \( y = 3x - 8 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 3 \) and the y - intercept \( b=-8 \). To graph this line, we start by plotting the y - intercept at the point \( (0,-8) \). Then, using the slope (rise over run), since the slope is 3 (or \( \frac{3}{1} \)), we can find another point. From \( (0,-8) \), we move up 3 units and to the right 1 unit, which gives us the point \( (1,-5) \). We can draw a line through these two points.

Step2: Analyze \( y=-2x - 3 \)

This is also in slope - intercept form with slope \( m=-2 \) and y - intercept \( b = - 3 \). We first plot the y - intercept at \( (0,-3) \). Using the slope of - 2 (or \( \frac{-2}{1} \)), from the point \( (0,-3) \), we move down 2 units and to the right 1 unit to get the point \( (1,-5) \).

Step3: Find the intersection

The two lines intersect at the point where their \( x \) and \( y \) values are equal. We can also solve the system algebraically to confirm the intersection point. Set \( 3x-8=-2x - 3 \). Add \( 2x \) to both sides: \( 3x + 2x-8=-2x+2x - 3 \), which simplifies to \( 5x-8=-3 \). Then add 8 to both sides: \( 5x-8 + 8=-3 + 8 \), so \( 5x = 5 \). Divide both sides by 5: \( x = 1 \). Substitute \( x = 1 \) into \( y = 3x-8 \), we get \( y=3(1)-8=3 - 8=-5 \). So the solution to the system of equations is the point of intersection of the two lines, which is \( (1,-5) \).

Answer:

The solution to the system of equations \(

$$\begin{cases}y = 3x-8\\y=-2x - 3\end{cases}$$

\) is \( x = 1,y=-5 \) (or the point \( (1,-5) \)).