Question

problems 8-10: determine whether each system of equations has one solution, no solution, or infinitely many solutions.
8.
\

$$\begin{cases} y = x + 4 y = x + 4 \\end{cases}$$

9.
\

$$\begin{cases} y = \\frac{4}{5}x + 7 y = \\frac{4}{5}x - 2 \\end{cases}$$

10.
\

$$\begin{cases} y = 2x + \\frac{1}{5} y = 2x + 42 \\end{cases}$$

Explanation:

Step1: Analyze System 8

The system is

$$\begin{cases} y = x + 4 \\ y = x + 4 \end{cases}$$

. Both equations are identical, meaning every point on the line satisfies both equations.

Step2: Analyze System 9

The system is

$$\begin{cases} y = -\frac{4}{5}x + 7 \\ y = \frac{4}{5}x - 2 \end{cases}$$

. The slopes $-\frac{4}{5}$ and $\frac{4}{5}$ are different, so the lines intersect at exactly one point.

Step3: Analyze System 10

The system is

$$\begin{cases} y = 2x + \frac{1}{5} \\ y = 2x + 42 \end{cases}$$

. The slopes are equal ($2=2$) but the y-intercepts $\frac{1}{5}$ and $42$ are different, so the lines are parallel and never intersect.

Answer:

1. System 8: Infinitely many solutions
2. System 9: One solution
3. System 10: No solution