Question

Susie determined that the system of equations below has infinitely many solutions.

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

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

Is she correct? *Use your Desmos calculator

(1 point)
Responses

No, the lines are parallel.
No, the lines are parallel.

Yes, Susie is correct.
Yes, Susie is correct.

No, there is no point of intersection.
No, there is no point of intersection.

She may or may not be correct as the lines possibly cross.

Answer 1

To determine whether Susie is correct and if the system of equations has infinitely many solutions, we need to find the equations of the lines given their points.

**Line 1** passes through the points (2, -13) and (-2, -1).

1. Calculate the slope of Line 1:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-13)}{-2 - 2} = \frac{12}{-4} = -3
\]

2. Use point-slope form \(y - y_1 = m(x - x_1)\) to find the equation of Line 1 using point (2, -13):

\[
y - (-13) = -3(x - 2) \\
y + 13 = -3x + 6 \\
y = -3x - 7
\]

**Line 2** passes through the points (5, -22) and (-3, 2).

1. Calculate the slope of Line 2:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-22)}{-3 - 5} = \frac{24}{-8} = -3
\]

2. Use point-slope form \(y - y_1 = m(x - x_1)\) to find the equation of Line 2 using point (5, -22):

\[
y - (-22) = -3(x - 5) \\
y + 22 = -3x + 15 \\
y = -3x - 7
\]

Both lines have the same slope and the same y-intercept, which means they are actually the same line, not just parallel lines. Hence, they have infinitely many solutions.

**Conclusion:** Yes, Susie is correct. The answer is:

**Yes, Susie is correct.**