Question

ine P passes through the points (2, 9) and (4, 13). Line Q passes through the points (1, -13) and (3, -19).

Select a number for each blank to make the sentence true.

Line P and Line Q intersect at the point (x,y)
, where x=


and y=


.

Answer 1

To find the intersection of lines P and Q, we need to start by determining the equations of both lines.

### Step 1: Find the equation of Line P

Line P passes through points (2, 9) and (4, 13).

First, we calculate the slope \( m \) of line P:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 9}{4 - 2} = \frac{4}{2} = 2
\]

Using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), we can use point (2, 9):

\[
y - 9 = 2(x - 2)
\]

Expanding this:

\[
y - 9 = 2x - 4 \\
y = 2x + 5
\]

### Step 2: Find the equation of Line Q

Line Q passes through points (1, -13) and (3, -19).

Calculating the slope \( m \) of line Q:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-19 - (-13)}{3 - 1} = \frac{-19 + 13}{2} = \frac{-6}{2} = -3
\]

Using point-slope form with point (1, -13):

\[
y - (-13) = -3(x - 1)
\]

Expanding this:

\[
y + 13 = -3x + 3 \\
y = -3x - 10
\]

### Step 3: Finding the intersection of Line P and Line Q

Now we have the equations of the lines:
- Line P: \( y = 2x + 5 \)
- Line Q: \( y = -3x - 10 \)

To find the point of intersection, set the equations equal to each other:

\[
2x + 5 = -3x - 10
\]

Solving for \( x \):

\[
2x + 3x = -10 - 5 \\
5x = -15 \\
x = -3
\]

Now, substitute \( x = -3 \) back into one of the line equations to find \( y \). We can use Line P:

\[
y = 2(-3) + 5 \\
y = -6 + 5 \\
y = -1
\]

### Conclusion

The lines intersect at the point \((-3, -1)\).

Thus, we have:

- \( x = -3 \)
- \( y = -1 \)

So, the complete sentence is:

Line P and Line Q intersect at the point (x,y), where \( x = -3 \) and \( y = -1 \).