Question

solve the system of linear equations by graphing.$y = -x + 3LXB0y = -\frac{5}{6}x - 2$show hintsolve the system of linear equations by graphing.$y = -\frac{1}{4}x + 7$$3x - 2y = 0$which ordered pair is a solution to the system of linear equations?

Explanation:

Step1: Solve first system algebraically

Set $-x+3 = -\frac{4}{5}x+4$
Multiply all terms by 5: $-5x+15 = -4x+20$
Rearrange: $-5x+4x = 20-15$ → $-x=5$ → $x=-5$
Substitute $x=-5$ into $y=-x+3$: $y=-(-5)+3=8$

Step2: Analyze Sophie's conclusion

Check slopes: $y=-\frac{1}{3}x+4$ has slope $-\frac{1}{3}$; $y=-\frac{5}{6}x-2$ has slope $-\frac{5}{6}$. Slopes are not equal, so lines intersect. Solve:
Set $-\frac{1}{3}x+4 = -\frac{5}{6}x-2$
Multiply by 6: $-2x+24 = -5x-12$
Rearrange: $-2x+5x = -12-24$ → $3x=-36$ → $x=-12$
Substitute $x=-12$ into $y=-\frac{1}{3}x+4$: $y=-\frac{1}{3}(-12)+4=4+4=8$
Sophie is wrong; lines intersect at $(-12,8)$.

Step3: Solve third system algebraically

Rewrite $3x-2y=0$ as $y=\frac{3}{2}x$
Set $\frac{3}{2}x = -\frac{1}{4}x+7$
Multiply by 4: $6x = -x+28$
Rearrange: $6x+x=28$ → $7x=28$ → $x=4$
Substitute $x=4$ into $y=\frac{3}{2}x$: $y=\frac{3}{2}(4)=6$

Answer:

1. Solution to first system: $(-5, 8)$
2. Sophie is incorrect; the system has a solution at $(-12, 8)$ (lines intersect because slopes are not equal)
3. Solution to third system: $(4, 6)$