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 \( y = 3x - 8 \)

Find two points. When \( x = 0 \), \( y = 3(0) - 8 = -8 \), so point \((0, -8)\). When \( x = 2 \), \( y = 3(2) - 8 = -2 \), so point \((2, -2)\). Plot these and draw the line.

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

Find two points. When \( x = 0 \), \( y = -2(0) - 3 = -3 \), so point \((0, -3)\). When \( x = 1 \), \( y = -2(1) - 3 = -5 \), so point \((1, -5)\). Plot these and draw the line.

Step3: Find Intersection

The lines intersect where \( 3x - 8 = -2x - 3 \). Solve: \( 3x + 2x = -3 + 8 \) → \( 5x = 5 \) → \( x = 1 \). Substitute \( x = 1 \) into \( y = 3x - 8 \): \( y = 3(1) - 8 = -5 \)? Wait, no, wait: \( 3x - 8 = -2x - 3 \) → \( 5x = 5 \) → \( x = 1 \). Then \( y = 3(1) - 8 = -5 \)? Wait, no, \( y = -2(1) - 3 = -5 \). Wait, earlier when \( x = 1 \) for \( y = -2x - 3 \), \( y = -5 \), and for \( y = 3x - 8 \), when \( x = 1 \), \( y = -5 \). Wait, but when \( x = 2 \) for \( y = 3x - 8 \), \( y = -2 \). Wait, maybe better to solve algebraically first. Set \( 3x - 8 = -2x - 3 \): \( 3x + 2x = -3 + 8 \) → \( 5x = 5 \) → \( x = 1 \), then \( y = 3(1) - 8 = -5 \). So the intersection is \((1, -5)\). But graphically, plot the two lines: \( y = 3x - 8 \) has slope 3, y-intercept -8; \( y = -2x - 3 \) has slope -2, y-intercept -3. The intersection is at \((1, -5)\).

Answer:

The solution (intersection point) is \((1, -5)\). To graph: plot \( y = 3x - 8 \) through \((0, -8)\) and \((2, -2)\), plot \( y = -2x - 3 \) through \((0, -3)\) and \((1, -5)\), and the lines intersect at \((1, -5)\).