Question

the two lines are parallel and the system has no solution.
the two lines are perpendicular and the system has infinitely many solutions.
the two lines are neither parallel nor perpendicular and the system has infinitely many solutions.
the two lines are parallel and the system has one solution.
the two lines are neither parallel nor perpendicular and the system has one solution.
the two lines are perpendicular and the system has one solution.

Explanation:

1. Parallel Check: Parallel lines have the same slope. The two lines here have different slopes (visible from different steepness and distinct points), so not parallel.
2. Perpendicular Check: Perpendicular lines have slopes that are negative reciprocals (product = -1). These lines don’t satisfy that, so not perpendicular.
3. Solution Type: A system of linear equations has infinitely many solutions only if the lines are coincident (same line). Here, the lines are distinct (different points), so they intersect at one point (one solution). Wait, correction: Wait, the initial marked option was wrong. Let's re - evaluate. Wait, the two lines: let's find their slopes. Blue line: passes through (3, 0) and (0, - 4). Slope $m_1=\frac{0 - (-4)}{3 - 0}=\frac{4}{3}$. Red line: passes through (3, - 1) and (0, - 5). Slope $m_2=\frac{-1-(-5)}{3 - 0}=\frac{4}{3}$? Wait, no, maybe I misread the points. Wait, the blue dot is at (3, 0)? Wait, the x - axis: the blue dot is at x = 3? Wait, the grid: x from - 10 to 10, y from - 10 to 10. Wait, the blue line: let's take two points. Blue line: (3, 0) and (0, - 4). Red line: (3, - 1) and (0, - 5). Wait, slope of blue: $\frac{0 - (-4)}{3-0}=\frac{4}{3}$. Slope of red: $\frac{-1-(-5)}{3 - 0}=\frac{4}{3}$. Wait, so they have the same slope? But the y - intercepts: blue has y - intercept - 4, red has y - intercept - 5. So they are parallel (same slope, different y - intercepts), so no solution. Wait, the options: "The two lines are parallel and the system has no solution" is an option. Let's re - check the options:

Option 1: "The two lines are parallel and the system has no solution."

Let's confirm slope:

For blue line: points (3, 0) and (0, - 4). Slope $m=\frac{0 - (-4)}{3-0}=\frac{4}{3}$.

For red line: points (3, - 1) and (0, - 5). Slope $m=\frac{-1-(-5)}{3 - 0}=\frac{4}{3}$.

Same slope, different y - intercepts (blue: - 4, red: - 5), so parallel, no solution.

The other options:

• Perpendicular? No, same slope.

• Infinitely many solutions? Only if same line (same slope and same y - intercept), not here.

• One solution? Only if they intersect, but parallel lines don't intersect.

So the correct option is "The two lines are parallel and the system has no solution."

Answer:

The two lines are parallel and the system has no solution.