Question

solve the system of linear equations by graphing.
$y = \frac{3}{2}x - 2$
$y = -\frac{1}{4}x + 5$
the solution is (\square, \square).
graph with coordinate plane and grid, options for line, undo, redo, reset

Explanation:

Step1: Find intercepts for \( y = \frac{3}{2}x - 2 \)

• When \( x = 0 \), \( y = -2 \), so the y - intercept is \( (0, -2) \).
• When \( y = 0 \), \( 0=\frac{3}{2}x - 2 \), solving for \( x \): \( \frac{3}{2}x=2 \), \( x=\frac{4}{3}\approx1.33 \), so the x - intercept is \( (\frac{4}{3}, 0) \).

Step2: Find intercepts for \( y = -\frac{1}{4}x + 5 \)

• When \( x = 0 \), \( y = 5 \), so the y - intercept is \( (0, 5) \).
• When \( y = 0 \), \( 0 = -\frac{1}{4}x+5 \), solving for \( x \): \( \frac{1}{4}x = 5 \), \( x = 20 \), so the x - intercept is \( (20, 0) \).

Step3: Graph the lines

Plot the intercepts for each line and draw the lines. The point where the two lines intersect is the solution to the system. By graphing (either by hand or using the grid provided), we can see that the two lines intersect at \( x = 4 \) and \( y = 4 \). We can also verify this algebraically by setting the two equations equal to each other:
\( \frac{3}{2}x-2=-\frac{1}{4}x + 5 \)
Multiply through by 4 to clear the fractions: \( 6x-8=-x + 20 \)
Add \( x \) to both sides: \( 7x-8 = 20 \)
Add 8 to both sides: \( 7x=28 \)
Divide by 7: \( x = 4 \)
Substitute \( x = 4 \) into \( y=\frac{3}{2}x-2 \): \( y=\frac{3}{2}(4)-2=6 - 2=4 \)

Answer:

\( (4, 4) \)