Question

find an equation for the line that passes through the points (2, 2) and (-6, 4).

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1)=(2, 2)\) and \((x_2, y_2)=(-6, 4)\). Then \(m=\frac{4 - 2}{-6 - 2}=\frac{2}{-8}=-\frac{1}{4}\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). We can use the point \((2, 2)\) and \(m =-\frac{1}{4}\). Substitute these values into the formula: \(y - 2=-\frac{1}{4}(x - 2)\).

Step3: Simplify the equation to slope - intercept form (\(y=mx + b\))

Expand the right - hand side: \(y - 2=-\frac{1}{4}x+\frac{1}{2}\). Then add 2 to both sides. Since \(2=\frac{4}{2}\), we have \(y=-\frac{1}{4}x+\frac{1}{2}+\frac{4}{2}=-\frac{1}{4}x+\frac{5}{2}\). We can also write it in standard form \(x + 4y=10\) (by multiplying both sides of \(y=-\frac{1}{4}x+\frac{5}{2}\) by 4: \(4y=-x + 10\), then \(x + 4y=10\)).

Answer:

The equation of the line is \(y =-\frac{1}{4}x+\frac{5}{2}\) (or \(x + 4y=10\))