To find the equation of the line that passes through the points (2, -1) and (8, 4), we will follow the steps below.
a. Write an equation for the line in point-slope form.
-
Calculate the slope (m) of the line: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{8 - 2} = \frac{4 + 1}{8 - 2} = \frac{5}{6} \]
-
Use the point-slope form of the line equation: The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Using point (2, -1): \[ y - (-1) = \frac{5}{6}(x - 2) \] Simplifying this gives: \[ y + 1 = \frac{5}{6}(x - 2) \] Thus, the equation in point-slope form is: \[ y + 1 = \frac{5}{6}(x - 2) \]
b. Rewrite the equation in standard form using integers.
-
Distributing and rearranging: Starting from the point-slope form: \[ y + 1 = \frac{5}{6}(x - 2) \]
Distribute: \[ y + 1 = \frac{5}{6}x - \frac{5}{3} \]
Subtract 1 from both sides: \[ y = \frac{5}{6}x - \frac{5}{3} - 1 \]
Convert -1 to a fraction with a common denominator (-3): \[ y = \frac{5}{6}x - \frac{5}{3} - \frac{3}{3} \] This simplifies to: \[ y = \frac{5}{6}x - \frac{8}{3} \]
-
Clear the fractions by multiplying the entire equation by 6 (the denominator): \[ 6y = 5x - 16 \]
-
Rearranging to standard form (Ax + By = C): To get it into standard form, bring \(5x\) to the left side: \[ -5x + 6y = -16 \]
To convert to the traditional standard form (with A as a positive integer), multiply through by -1: \[ 5x - 6y = 16 \]
Thus, the final equations are:
- Point-slope form: \( y + 1 = \frac{5}{6}(x - 2) \)
- Standard form: \( 5x - 6y = 16 \)