Question

Susie determined that the system of equations below has infinitely many solutions. *Use your graphing calculator!

The first line goes through the points (2, -13) and (-2, -1)

The second line goes through the points (5, -22) and (-3, 2)

(1 point)

Answer 1

To determine whether the given system of equations has infinitely many solutions, we first need to find the equations of the two lines given their respective points.

**Finding the equation of the first line:**

The first line passes through the points \((2, -13)\) and \((-2, -1)\).

1. **Calculate the slope (m)** using the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (2, -13)\) and \((x_2, y_2) = (-2, -1)\):
\[
m = \frac{-1 - (-13)}{-2 - 2} = \frac{-1 + 13}{-4} = \frac{12}{-4} = -3
\]

2. **Use the point-slope form** of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Using point \((2, -13)\):
\[
y - (-13) = -3(x - 2)
\]
\[
y + 13 = -3x + 6
\]
\[
y = -3x + 6 - 13
\]
\[
y = -3x - 7
\]

**Finding the equation of the second line:**

The second line passes through the points \((5, -22)\) and \((-3, 2)\).

1. **Calculate the slope (m)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (5, -22)\) and \((x_2, y_2) = (-3, 2)\):
\[
m = \frac{2 - (-22)}{-3 - 5} = \frac{2 + 22}{-8} = \frac{24}{-8} = -3
\]

2. **Use the point-slope form**:
Using point \((5, -22)\):
\[
y - (-22) = -3(x - 5)
\]
\[
y + 22 = -3x + 15
\]
\[
y = -3x + 15 - 22
\]
\[
y = -3x - 7
\]

**Conclusion:**

Both lines share the same equation:
\[
y = -3x - 7
\]
Since the two lines are identical, this means they have infinitely many solutions.

Thus, Susie was correct in determining that the system of equations has infinitely many solutions.