Question

Determine the number of solutions to the system of linear equations shown on the graph.

coordinate plane with one line that passes through the points 0 comma 2 and 1 comma 0 and another line that passes through the points 0 comma negative 1 and 1 comma negative 2
One solution at (−4, 3)
One solution at (3, −4)
No solution
Infinitely many solutions

Answer 1

To determine the number of solutions to the system of linear equations represented by the two lines, we need to analyze their slopes and their intercepts based on the given points.

**First Line:**
- It passes through the points (0, 2) and (1, 0).
- The slope (m) of the line can be calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{1 - 0} = \frac{-2}{1} = -2
\]
- The y-intercept \(b\) is at (0, 2), so the equation of the first line is:
\[
y = -2x + 2
\]

**Second Line:**
- It passes through the points (0, -1) and (1, -2).
- The slope of the line can be calculated as:
\[
m = \frac{-2 - (-1)}{1 - 0} = \frac{-2 + 1}{1} = \frac{-1}{1} = -1
\]
- The y-intercept \(b\) is at (0, -1), so the equation of the second line is:
\[
y = -x - 1
\]

**Finding the Point of Intersection:**
To find the number of solutions to the system, we can find the point where the two lines intersect by setting their equations equal to each other:
\[
-2x + 2 = -x - 1
\]
Now, solve for \(x\):
1. Rearranging gives:
\[
-2x + x = -1 - 2
\]
\[
-x = -3
\]
\[
x = 3
\]

2. Substitute \(x = 3\) back into one of the equations to find \(y\):
Using \(y = -x - 1\):
\[
y = -3 - 1 = -4
\]

So the lines intersect at the point \( (3, -4) \).

**Conclusions on Solutions:**
1. The lines intersect at a single point, which means there is **exactly one solution** to the system of equations.

Therefore, the final answer is:
**One solution at (3, -4).**