Question

do you know how? solve each system by graphing. tell whether the system has one solution, infinitely many solutions, or no solution. 1. $y = x - 1$ $y = -3x - 5$ 2. $y = \frac{4}{3}x - 2$ $3y - 4x = -6$ 3. $y = 3x - 4$ $y - 3x = 1$

Explanation:

Step1: Solve system 1 algebraically first

Set $x-1=-3x-5$
$x+3x=-5+1$
$4x=-4$
$x=-1$
Substitute $x=-1$ into $y=x-1$: $y=-1-1=-2$
Lines have different slopes ($1$ and $-3$), so they intersect once.

Step2: Analyze system 2

Rewrite second equation: $3y=4x-6 \implies y=\frac{4}{3}x-2$
This matches the first equation, so lines are identical.

Step3: Analyze system 3

Rewrite second equation: $y=3x+1$
First line: $y=3x-4$. Same slope ($3$), different y-intercepts, so lines are parallel.

Answer:

1. One solution: $(-1, -2)$
2. Infinitely many solutions
3. No solution