Question

solve the system of linear equations by graphing. (3 pts)
$y = \frac{1}{3}x + 2$
$y = 2x - 3$

Explanation:

Step1: Find two points for \( y = \frac{1}{3}x + 2 \)

• When \( x = 0 \), \( y=\frac{1}{3}(0)+2 = 2 \), so the point is \( (0, 2) \).
• When \( x = 3 \), \( y=\frac{1}{3}(3)+2 = 1 + 2 = 3 \), so the point is \( (3, 3) \).

Step2: Find two points for \( y = 2x - 3 \)

• When \( x = 0 \), \( y = 2(0)-3=-3 \), so the point is \( (0, -3) \).
• When \( x = 2 \), \( y = 2(2)-3 = 4 - 3 = 1 \), so the point is \( (2, 1) \).

Step3: Graph the lines

• Plot the points for each line and draw the lines. The intersection point of the two lines is the solution of the system. By graphing, we can see that the lines intersect at \( x = 3 \) and \( y = 3 \) (we can also solve algebraically: set \( \frac{1}{3}x + 2 = 2x - 3 \), \( 2 + 3 = 2x-\frac{1}{3}x \), \( 5=\frac{5}{3}x \), \( x = 3 \), then \( y = 2(3)-3 = 3 \)).

Answer:

The solution to the system is \( x = 3 \), \( y = 3 \) or the ordered pair \( (3, 3) \).