Question

solving systems by graphinguse your knowledge of solving systems by graphing to answer each question below.1. arithel graphed a system of equations with infinitely many solutions. what must be true about the equations of the lines?a. they have the same slope.b. they have different y-intercepts.c. the lines are parallel.d. all of the above.2. willa graphed a system of equations on the same coordinate grid. the equations had the same slope, but the y-intercepts were different. which must be true?a. the system has infinitely many solutions.b. the system has one solution.c. the system has no solution.d. there is not enough information to tell.solve each system of equations below by graphing.3. $y = -\frac{3}{2}x + 3LXB05x + 5y = 10$solution:5. $y = \frac{3}{2}x - 2LXB1x + 2y = 8$solution:

Explanation:

Step1: Analyze Question 1

A system with infinitely many solutions means the lines are identical, so they have the same slope and same y-intercepts (ruling out b, c, d).

Step2: Analyze Question 2

Lines with same slope but different y-intercepts are parallel, never intersecting, so no solution.

Step3: Solve System 3

Set equations equal: $\frac{3}{2}x + 3 = 2x - 6$
Rearrange: $3 + 6 = 2x - \frac{3}{2}x$
Simplify: $9 = \frac{1}{2}x \implies x=18$
Substitute $x=18$: $y=2(18)-6=30$

Step4: Solve System 4

Rewrite $5x+5y=10$ as $y=-x+2$.
Set equal to $y=-x+7$: $-x+7=-x+2 \implies 7=2$ (false, parallel lines, no solution)

Step5: Solve System 5 (assumed: $y=\frac{3}{2}x-2$ and $y=\frac{3}{2}x+3$)

Same slope, different y-intercepts: parallel lines, no solution

Step6: Solve System 6

Rewrite $x+2y=8$ as $y=-\frac{1}{2}x+4$.
Set equal to $y=\frac{1}{2}x+4$: $\frac{1}{2}x+4=-\frac{1}{2}x+4$
Simplify: $x=0$
Substitute $x=0$: $y=4$

Answer:

1. a. They have the same slope.
2. c. The system has no solution.
3. $(18, 30)$
4. No solution
5. No solution
6. $(0, 4)$