Let's go through each step systematically.
1. Use slope formula to find the slope.
The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((6, -2)\) and \((-2, 4)\):
\[ m = \frac{4 - (-2)}{-2 - 6} = \frac{4 + 2}{-2 - 6} = \frac{6}{-8} = -\frac{3}{4} \]
2. Substitute a point and slope in point-slope form.
We'll use the point \((-2, 4)\) and the slope \(-\frac{3}{4}\). The point-slope form of the line equation is: \[ y - y_1 = m(x - x_1) \] Substituting in the values: \[ y - 4 = -\frac{3}{4}(x - (-2)) \quad \text{or} \quad y - 4 = -\frac{3}{4}(x + 2) \]
3. Distribute the slope through the parentheses.
Now distribute \(-\frac{3}{4}\): \[ y - 4 = -\frac{3}{4}x - \frac{3}{4} \cdot 2 \] Calculating \(-\frac{3}{4} \cdot 2\): \[ -\frac{3}{4} \cdot 2 = -\frac{6}{4} = -\frac{3}{2} \] So we have: \[ y - 4 = -\frac{3}{4}x - \frac{3}{2} \]
4. Solve for the y-variable.
Now, add \(4\) to both sides: \[ y = -\frac{3}{4}x - \frac{3}{2} + 4 \] Convert \(4\) to a fraction with a common denominator: \[ 4 = \frac{8}{2} \] Now combine the constants: \[ y = -\frac{3}{4}x - \frac{3}{2} + \frac{8}{2} = -\frac{3}{4}x + \frac{5}{2} \]
Final Output
So, the final equation in slope-intercept form is: \[ y = -\frac{3}{4}x + \frac{5}{2} \]