Question

solve the following system of equations graphically on the set of axes below.
$y = -\frac{1}{2}x - 8$
$2x - y = 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = -\frac{1}{2}x - 8 \)

This is in slope - intercept form \( y=mx + b \), where the slope \( m=-\frac{1}{2} \) and the y - intercept \( b=-8 \). To find two points on this line:

• When \( x = 0 \), \( y=-\frac{1}{2}(0)-8=-8 \). So one point is \( (0, - 8) \).
• When \( x = - 2 \), \( y=-\frac{1}{2}(-2)-8=1 - 8=-7 \). So another point is \( (-2,-7) \).

Step2: Analyze the second equation \( 2x - y=3 \), rewrite it in slope - intercept form

Solve for \( y \): \( y = 2x-3 \). Here, the slope \( m = 2 \) and the y - intercept \( b=-3 \). To find two points on this line:

• When \( x = 0 \), \( y=2(0)-3=-3 \). So one point is \( (0,-3) \).
• When \( x = 1 \), \( y=2(1)-3=-1 \). So another point is \( (1,-1) \).

Step3: Find the intersection point (graphically)

When we plot the line \( y = -\frac{1}{2}x - 8 \) (using points like \( (0, - 8) \) and \( (-2,-7) \)) and the line \( y = 2x-3 \) (using points like \( (0,-3) \) and \( (1,-1) \)), we can see that the two lines intersect at the point \( (-2,-9) \). We can verify this algebraically as well:
Substitute \( y = -\frac{1}{2}x - 8 \) into \( 2x - y=3 \):
\( 2x-(-\frac{1}{2}x - 8)=3 \)
\( 2x+\frac{1}{2}x + 8=3 \)
\( \frac{4x + x}{2}=3 - 8=-5 \)
\( \frac{5x}{2}=-5 \)
\( 5x=-10 \)
\( x=-2 \)
Substitute \( x = - 2 \) into \( y = 2x-3 \): \( y=2(-2)-3=-4 - 3=-7 \)? Wait, no, substitute into \( y = -\frac{1}{2}x - 8 \): \( y=-\frac{1}{2}(-2)-8 = 1-8=-7 \)? Wait, there was a mistake in the graphical analysis earlier. Let's do the substitution correctly.

Substitute \( y=-\frac{1}{2}x - 8 \) into \( 2x - y=3 \):

\( 2x-(-\frac{1}{2}x - 8)=3 \)

\( 2x+\frac{1}{2}x+8 = 3 \)

\( \frac{4x + x}{2}=3 - 8\)

\( \frac{5x}{2}=-5\)

Multiply both sides by 2: \( 5x=-10\)

\( x = - 2\)

Now substitute \( x=-2 \) into \( y = -\frac{1}{2}x - 8 \):

\( y=-\frac{1}{2}(-2)-8=1 - 8=-7\)

Wait, let's check with the second equation \( y = 2x-3 \):

\( y=2(-2)-3=-4 - 3=-7\). So the intersection point is \( (-2,-7) \). (The earlier graphical analysis had an error in finding the intersection point. Let's re - check the points for the first line: when \( x=-2 \), \( y = -\frac{1}{2}(-2)-8=1 - 8=-7 \), correct. For the second line, when \( x=-2 \), \( y=2(-2)-3=-4 - 3=-7 \). So the two lines intersect at \( (-2,-7) \))

Answer:

The solution to the system of equations is \( x=-2,y = - 7 \) or the ordered pair \( (-2,-7) \)