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 3 and 1 comma 0 and another line that passes through the points negative 3 comma 2 and 0 comma negative 1

No solution
Infinitely many solutions
One solution at (−3, 2)
One solution at (2, −3)

Answer 1

To determine the number of solutions to the system of linear equations represented by the lines described, we first need to analyze their slopes and intercepts to see if they are parallel, coincident, or intersecting.

1. **Line passing through (0, 3) and (1, 0)**:
- The slope (m) of the line can be calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{1 - 0} = \frac{-3}{1} = -3
\]
- The equation of the line in slope-intercept form (y = mx + b) is:
- The y-intercept (b) is 3, so the equation is:
\[
y = -3x + 3
\]

2. **Line passing through (-3, 2) and (0, -1)**:
- The slope (m) of this line is:
\[
m = \frac{-1 - 2}{0 - (-3)} = \frac{-3}{3} = -1
\]
- The equation of the line can be derived by calculating the y-intercept as follows:
- Using the point (0, -1), we know b = -1. So the equation is:
\[
y = -x - 1
\]

3. **Comparing the two lines**:
- The first line's equation is \(y = -3x + 3\).
- The second line's equation is \(y = -x - 1\).

4. **Finding the intersection**:
- Set the equations equal to find the intersection:
\[
-3x + 3 = -x - 1
\]
\[
-3x + x = -1 - 3
\]
\[
-2x = -4
\]
\[
x = 2
\]
- Substitute \(x = 2\) back into one of the equations to solve for \(y\):
\[
y = -3(2) + 3 = -6 + 3 = -3
\]

Thus, the two lines intersect at the point \((2, -3)\).

5. **Final Analysis**:
Since the lines are not parallel and from our calculations, they intersect at a single point, we conclude the system has one unique solution.

The correct answer is:
**One solution at (2, -3)**.