Question

solve the following system of equations graphically on the set of axes below.
$y = -dfrac{1}{2}x + 7$
$y = dfrac{3}{2}x - 5$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = -\frac{1}{2}x + 7 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = -\frac{1}{2} \) and the y - intercept \( b = 7 \). To plot this line, we can start by marking the y - intercept at the point \( (0,7) \). Then, using the slope (rise over run), from the point \( (0,7) \), we can go down 1 unit (since the slope is negative) and to the right 2 units to get the next point. For example, from \( (0,7) \), moving down 1 (change in y) and right 2 (change in x) gives us the point \( (2,6) \).

Step2: Analyze the second equation \( y=\frac{3}{2}x - 5 \)

This is also in slope - intercept form with slope \( m=\frac{3}{2} \) and y - intercept \( b=- 5 \). We mark the y - intercept at the point \( (0, - 5) \). Then, using the slope, from the point \( (0,-5) \), we go up 3 units (change in y) and to the right 2 units (change in x) to get the next point. For example, from \( (0,-5) \), moving up 3 and right 2 gives us the point \( (2,-2) \).

Step3: Find the intersection point (graphical solution)

When we plot both lines, the point where they intersect is the solution to the system of equations. To find the intersection algebraically (to confirm the graphical solution), we set the two equations equal to each other:
\[
-\frac{1}{2}x + 7=\frac{3}{2}x-5
\]
Add \( \frac{1}{2}x \) to both sides:
\[
7=\frac{3}{2}x+\frac{1}{2}x - 5
\]
Simplify the right - hand side: \( \frac{3}{2}x+\frac{1}{2}x=2x \), so we have \( 7 = 2x-5 \)
Add 5 to both sides: \( 7 + 5=2x \), so \( 12 = 2x \)
Divide both sides by 2: \( x = 6 \)
Substitute \( x = 6 \) into the first equation \( y=-\frac{1}{2}(6)+7=-3 + 7 = 4 \)

Answer:

The solution to the system of equations (the point of intersection of the two lines) is \( (6,4) \)