Question

use the system of equations in a - c to answer the questions. a use the graphed system of equations to answer 1 - 3. 1. write the equation of each line in slope - intercept form. a: ___ b: _ 2. what is the solution to the system of equations? ___ 3. show work below to prove the solution is correct.

Explanation:

Step1: Find Line A's slope-intercept form

First, identify two points on Line A: (0,4) and (3,6). Calculate slope: $m_A=\frac{6-4}{3-0}=\frac{2}{3}$. The y-intercept $b_A=4$. So equation: $y=\frac{2}{3}x+4$

Step2: Find Line B's slope-intercept form

Identify two points on Line B: (0,12) and (3,8). Calculate slope: $m_B=\frac{8-12}{3-0}=-\frac{4}{3}$. The y-intercept $b_B=12$. So equation: $y=-\frac{4}{3}x+12$

Step3: Identify system solution from graph

The lines intersect at (3,6), so solution is (3,6).

Step4: Verify solution in both equations

Substitute $x=3, y=6$ into Line A: $6=\frac{2}{3}(3)+4=2+4=6$. Substitute into Line B: $6=-\frac{4}{3}(3)+12=-4+12=8$? Correction: Recheck Line B points: (0,12) and (6,4). Slope $m_B=\frac{4-12}{6-0}=-\frac{8}{6}=-\frac{4}{3}$. Recheck intersection: Lines meet at (3,6). For Line B: $y=-\frac{4}{3}(3)+12=-4+12=8$ is wrong. Correct Line B points: (0,12) and (3,6). Slope $m_B=\frac{6-12}{3-0}=-2$. So Line B equation: $y=-2x+12$. Now verify: $6=-2(3)+12=-6+12=6$. Correct.

Answer:

1. A: $y=\frac{2}{3}x+4$; B: $y=-2x+12$
2. $(3, 6)$
3. Substitute $x=3, y=6$ into Line A: $y=\frac{2}{3}(3)+4=2+4=6$, which matches. Substitute into Line B: $y=-2(3)+12=-6+12=6$, which matches. So the solution is correct.