Question

skill #5: i can solve systems of linear equations by graphing. solve each system by graphing. a) $y = -\frac{1}{3}x + 2$ $y = -2x - 3$

Explanation:

Step1: Analyze the first equation \( y = -\frac{1}{3}x + 2 \)

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For \( y = -\frac{1}{3}x + 2 \), the y-intercept \( b = 2 \), so the line passes through \( (0, 2) \). The slope \( m = -\frac{1}{3} \), which means from the y-intercept, we can go down 1 unit and right 3 units to get another point, e.g., \( (3, 1) \), or up 1 unit and left 3 units to get \( (-3, 3) \).

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

For \( y = -2x - 3 \), the y-intercept \( b = -3 \), so the line passes through \( (0, -3) \). The slope \( m = -2 \), which means from the y-intercept, we can go down 2 units and right 1 unit to get \( (1, -5) \), or up 2 units and left 1 unit to get \( (-1, -1) \).

Step3: Graph the two lines

Plot the points for each line and draw the lines. The first line ( \( y = -\frac{1}{3}x + 2 \)) passes through \( (0, 2) \), \( (3, 1) \), \( (-3, 3) \). The second line ( \( y = -2x - 3 \)) passes through \( (0, -3) \), \( (1, -5) \), \( (-1, -1) \).

Step4: Find the intersection point

The solution to the system of linear equations is the point where the two lines intersect. By graphing, we can see that the two lines intersect at \( x = -3 \) and \( y = 3 \)? Wait, no, let's solve it algebraically to check. Set \( -\frac{1}{3}x + 2 = -2x - 3 \). Multiply both sides by 3 to eliminate the fraction: \( -x + 6 = -6x - 9 \). Add \( 6x \) to both sides: \( 5x + 6 = -9 \). Subtract 6: \( 5x = -15 \). Divide by 5: \( x = -3 \). Then substitute \( x = -3 \) into \( y = -2x - 3 \): \( y = -2(-3) - 3 = 6 - 3 = 3 \). Wait, but when we graph \( y = -\frac{1}{3}x + 2 \) at \( x = -3 \), \( y = -\frac{1}{3}(-3) + 2 = 1 + 2 = 3 \). So the intersection point is \( (-3, 3) \). Wait, but let's check the graph again. Wait, maybe I made a mistake in the graphing points. Wait, for \( y = -2x - 3 \), when \( x = -3 \), \( y = -2(-3) - 3 = 6 - 3 = 3 \). For \( y = -\frac{1}{3}x + 2 \), when \( x = -3 \), \( y = 1 + 2 = 3 \). So the intersection is at \( (-3, 3) \). Wait, but let's check the slope of the first line: \( -\frac{1}{3} \), so from \( (0, 2) \), going left 3 (x=-3) gives y=3, which is correct. The second line: slope -2, from \( (0, -3) \), going left 1 (x=-1) gives y=-1, left 2 (x=-2) gives y=1, left 3 (x=-3) gives y=3. So yes, they intersect at (-3, 3).

Answer:

The solution to the system is \( x = -3 \), \( y = 3 \), so the ordered pair is \( \boxed{(-3, 3)} \)