Question

which systems of equations have infinitely many solutions? select all that apply. click or tap the correct system of equations.

Explanation:

Step1: Recall infinite solution condition

A linear system $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$ has infinitely many solutions if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.

Step2: Check System 1

System: $y=2x-3$, $y=2x-4$
Rewrite: $2x-y=3$, $2x-y=4$
$\frac{2}{2}=\frac{-1}{-1}
eq\frac{3}{4}$ → No infinite solutions.

Step3: Check System 2

System: $y=3x-1$, $y=3x+2$
Rewrite: $3x-y=1$, $3x-y=-2$
$\frac{3}{3}=\frac{-1}{-1}
eq\frac{1}{-2}$ → No infinite solutions.

Step4: Check System 3

System: $2x+4y=8$, $y=-\frac{1}{2}x+2$
Rewrite second equation: $\frac{1}{2}x+y=2$ → $x+2y=4$
First equation: $2x+4y=8$ → divide by 2: $x+2y=4$
$\frac{2}{1}=\frac{4}{2}=\frac{8}{4}=2$ → Infinite solutions.

Step5: Check System 4

System: $\frac{1}{3}x+y=2$, $x+3y=6$
Multiply first equation by 3: $x+3y=6$
$\frac{\frac{1}{3}}{1}=\frac{1}{3}=\frac{2}{6}=\frac{1}{3}$ → Infinite solutions.

Answer:

• $2x+4y=8$ and $y=-\frac{1}{2}x+2$
• $\frac{1}{3}x+y=2$ and $x+3y=6$