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:
- \( 12x - 5y + 1 = 0 \)
- \( 5x + 3y - 44 = 0 \)
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).