Question

Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?

Answer 1

To determine whether the line through the points (3, 7) and (-2, -5) intersects with the line through the points (4, 8) and (10, -2), we need to find the equations of both lines and check if they intersect.

**Step 1: Find the slope and equation of the first line (through (3, 7) and (-2, -5))**

The slope \( m \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points (3, 7) and (-2, -5):
- \( (x_1, y_1) = (3, 7) \)
- \( (x_2, y_2) = (-2, -5) \)

Calculating the slope:
\[
m = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5}
\]

Now we can use the point-slope form of the line equation \( y - y_1 = m(x - x_1) \):
\[
y - 7 = \frac{12}{5}(x - 3)
\]
Multiply through by 5 to eliminate the fraction:
\[
5(y - 7) = 12(x - 3)
\]
\[
5y - 35 = 12x - 36
\]
Rearranging this gives:
\[
12x - 5y + 1 = 0 \quad \text{(Equation 1)}
\]

**Step 2: Find the slope and equation of the second line (through (4, 8) and (10, -2))**

Using the same slope formula for the points (4, 8) and (10, -2):
- \( (x_1, y_1) = (4, 8) \)
- \( (x_2, y_2) = (10, -2) \)

Calculating the slope:
\[
m = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3}
\]

Using the point-slope form:
\[
y - 8 = -\frac{5}{3}(x - 4)
\]
Multiply through by 3 to eliminate the fraction:
\[
3(y - 8) = -5(x - 4)
\]
\[
3y - 24 = -5x + 20
\]
Rearranging this gives:
\[
5x + 3y - 44 = 0 \quad \text{(Equation 2)}
\]

**Step 3: Solve the system of equations to find intersection**

Now we solve the equations:
1. \( 12x - 5y + 1 = 0 \)
2. \( 5x + 3y - 44 = 0 \)

From Equation 1, solve for \( y \):
\[
12x - 5y + 1 = 0 \implies 5y = 12x + 1 \implies y = \frac{12}{5}x + \frac{1}{5}
\]

Substituting this into Equation 2:
\[
5x + 3\left(\frac{12}{5}x + \frac{1}{5}\right) - 44 = 0
\]
Expanding this:
\[
5x + \frac{36}{5}x + \frac{3}{5} - 44 = 0
\]
Multiply through by 5 to eliminate fractions:
\[
25x + 36x + 3 - 220 = 0
\]
Combining like terms:
\[
61x - 217 = 0 \implies 61x = 217 \implies x = \frac{217}{61} = 3.56
\]

Now substitute \( x \) back into one of the original equations to find \( y \). Using the rearranged form from Equation 1:
\[
y = \frac{12}{5} \cdot \frac{217}{61} + \frac{1}{5} = \frac{12 \cdot 217 + 1}{5 \cdot 61}
\]
Calculating gives:
\[
y \approx 0.66
\]

**Conclusion**
The two lines do intersect. Thus, the answer is **Yes**, the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2).