Question

question
what is the equation of the line that passes through the point $(-8, 6)$ and has a slope of $-\frac{1}{4}$?

Explanation:

Step1: Recall point - slope form

The point - slope form of a linear equation is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.
Here, \(x_1=-8\), \(y_1 = 6\) and \(m=-\frac{1}{4}\).

Step2: Substitute values into point - slope form

Substitute \(x_1=-8\), \(y_1 = 6\) and \(m =-\frac{1}{4}\) into the point - slope formula:
\(y-6=-\frac{1}{4}(x - (-8))\)
Simplify the right - hand side: \(y - 6=-\frac{1}{4}(x + 8)\)

Step3: Convert to slope - intercept form (optional, but common)

Distribute the \(-\frac{1}{4}\) on the right - hand side:
\(y-6=-\frac{1}{4}x-2\)
Then, add 6 to both sides of the equation:
\(y=-\frac{1}{4}x-2 + 6\)
\(y=-\frac{1}{4}x + 4\)

We can also leave it in point - slope form: \(y - 6=-\frac{1}{4}(x + 8)\) or convert to standard form \(x+4y=16\) (by multiplying both sides of \(y=-\frac{1}{4}x + 4\) by 4: \(4y=-x + 16\), then \(x + 4y=16\)). But the most common forms are slope - intercept or point - slope. Let's present the slope - intercept form.

Answer:

The equation of the line is \(y=-\frac{1}{4}x + 4\) (or in point - slope form \(y - 6=-\frac{1}{4}(x + 8)\) or in standard form \(x + 4y=16\))