Question

13 solve the system of linear equations by graphing. (3 pts) $y = 6x - 5$ $y = 5x - 4$

Explanation:

Step1: Find intercepts for \( y = 6x - 5 \)

• When \( x = 0 \), \( y = 6(0) - 5 = -5 \). So one point is \( (0, -5) \).
• When \( y = 0 \), \( 0 = 6x - 5 \Rightarrow 6x = 5 \Rightarrow x = \frac{5}{6} \approx 0.83 \). Another point is \( (\frac{5}{6}, 0) \).

Step2: Find intercepts for \( y = 5x - 4 \)

• When \( x = 0 \), \( y = 5(0) - 4 = -4 \). So one point is \( (0, -4) \).
• When \( y = 0 \), \( 0 = 5x - 4 \Rightarrow 5x = 4 \Rightarrow x = \frac{4}{5} = 0.8 \). Another point is \( (\frac{4}{5}, 0) \).

Step3: Graph the lines

Plot the points for each line and draw the lines. The intersection point of the two lines is the solution. To find it algebraically (since graphing might be approximate), set \( 6x - 5 = 5x - 4 \).

Step4: Solve for \( x \)

\( 6x - 5 = 5x - 4 \)
Subtract \( 5x \) from both sides: \( x - 5 = -4 \)
Add 5 to both sides: \( x = 1 \)

Step5: Find \( y \)

Substitute \( x = 1 \) into \( y = 6x - 5 \): \( y = 6(1) - 5 = 1 \)

Answer:

The solution to the system is \( (1, 1) \) (the point where the two lines intersect).