Question

a line has a slope of $-\frac{1}{5}$ and passes through the point (10, 6). what is its equation in point - slope form? use the specified point in your equation. write your answer using integers, proper fractions, and improper fractions. simplify all fractions.

Explanation:

Step1: Recall point - slope formula

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \(m\) is the slope of the line and \((x_1,y_1)\) is a point on the line.
We are given that \(m =-\frac{1}{5}\) and the point \((x_1,y_1)=(10,6)\).

Step2: Substitute values into the formula

Substitute \(m =-\frac{1}{5}\), \(x_1 = 10\) and \(y_1=6\) into the point - slope formula:
\(y - 6=-\frac{1}{5}(x - 10)\)

Step3: Simplify the equation (optional, but to match the requirement of using integers etc.)

We can distribute the \(-\frac{1}{5}\) on the right - hand side:
\(y - 6=-\frac{1}{5}x+2\)
Then, add 6 to both sides of the equation:
\(y=-\frac{1}{5}x + 2+6\)
\(y=-\frac{1}{5}x+8\)
But if we want to keep it in point - slope form (as the question asks for point - slope form), the equation \(y - 6=-\frac{1}{5}(x - 10)\) is also correct. If we want to clear the fraction, we can multiply both sides by 5:
\(5(y - 6)=-1(x - 10)\)
\(5y-30=-x + 10\)
\(x+5y=40\) (this is the standard form, but the question asks for point - slope form first)

Answer:

The point - slope form of the line is \(y - 6=-\frac{1}{5}(x - 10)\) (or if we simplify further to slope - intercept form \(y=-\frac{1}{5}x + 8\) or standard form \(x + 5y=40\)). But based on the point - slope form requirement, the answer is \(y - 6=-\frac{1}{5}(x - 10)\)