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

Answer 1

To determine if the system of equations has infinitely many solutions, we need to check if the two lines are actually the same line. This can happen if they have the same slope and y-intercept, which means that they are identical.

### Step 1: Find the equations of both lines.

**For the first line:**

We can use the formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (2, -13) and (-2, -1):

\[
m = \frac{-1 - (-13)}{-2 - 2} = \frac{-1 + 13}{-4} = \frac{12}{-4} = -3
\]

Now we can use point-slope form with point (2, -13):
\[
y - y_1 = m(x - x_1)
\]
\[
y + 13 = -3(x - 2)
\]
\[
y + 13 = -3x + 6
\]
\[
y = -3x + 6 - 13
\]
\[
y = -3x - 7
\]

**Equation of the first line:** \( y = -3x - 7 \)

**For the second line:**

Using the points (5, -22) and (-3, 2):

\[
m = \frac{2 - (-22)}{-3 - 5} = \frac{2 + 22}{-8} = \frac{24}{-8} = -3
\]

Now we use the point-slope form with point (5, -22):
\[
y - y_1 = m(x - x_1)
\]
\[
y + 22 = -3(x - 5)
\]
\[
y + 22 = -3x + 15
\]
\[
y = -3x + 15 - 22
\]
\[
y = -3x - 7
\]

**Equation of the second line:** \( y = -3x - 7 \)

### Step 2: Compare both equations.

Both lines are represented by the same equation \( y = -3x - 7 \). This means that they are indeed the same line, which implies that the system of equations has infinitely many solutions.

### Conclusion:

Susie is correct that the system of equations has infinitely many solutions.