### Finding Slope from an Equation Examples and Worksheets

Get here our tutorial lesson, examples, and worksheets about finding slope from an equation. In our previous posts, we discussed finding a slope from a graph and from two given points.

When we are asked to find the slope of a line given two distinct points, it would be easy for us to do it, since all we have to do is assign one point to be the firs point and the other would be the second point, and then simply substitute the value accordingly to the formula. But, how will we find the slope of a line if the given is only its equation?

Basically, there are two ways we can consider in finding the slope of a line if the given is its equation. It is either by using the definition of the slope, wherein we need to find first two distinct points that lies in the line or we simply transform the equation into slope-intercept form, $$y=mx+b$$, wherein the value that replaces m is the slope. To differentiate between the two let us consider the following examples.

Examples

Find the slope of the line given the following equation.

1. $$4x-y=-8$$

SOLUTION:

a. By using the definition of slope, we set $$x=0$$ to get $$y$$.

\small \begin{align*}\displaystyle &\;\;\;4\left ( 0 \right )-y=-8 &&\textup{Substituting the value of}\; x \; \textup{to the equation}\\& \qquad -1\left ( -y=-8 \right )&&\textup{Multiplying -1 to make it positive} \\& \qquad \qquad \quad \therefore y=8 \end{align*}
Letting $$y=0$$, we are able to solve $$x$$ from the given equation.

\small \begin{align*}\displaystyle &\;\;\;4x-0=-8 &&\textup{Substituting the value of}\; y \; \textup{to the equation}\\& \qquad 4x=-8 \\& \qquad \frac{4x}{4}=\frac{-8}{4}&&\textup{Divide both sides by 4 to get the value of}\; x \\& \qquad \therefore x=-2 \end{align*}

The two points are $$\left(0,8 \right )$$ and $$\left(-2,0\right)$$. Thus,

$$\displaystyle m = \frac{y_2-y_1}{x_2-x_1}=\frac{0-8}{-2-0}=\frac{-8}{-2}=4.$$

b. Transforming the equation into slope-intercept form, $$y=mx+b$$.

\small \begin{align*}\displaystyle &\;\;\;4x-y=-8 \\& \qquad 4x-4x-y=-8-4x &&\textup{Subtract both sides with}\; 4x \\& \qquad -y=-8-4x\\& \qquad -y=-4x-8 \\& \qquad -1 \left ( -y=-8-4x \right )&&\textup{Multiply by -1 to make both sides positive}\; x \\& \qquad \therefore y=4x+8 \end{align*}

The slope is said to be the value that replaces m in the equation, thus 4 is the slope.

To summarize the procedures, in the first method, we need to find first the $$x$$ and $$y$$ intercepts which will serve as the two different points. Then, solve the slope by following the steps in finding the slope given two points. X-intercept refers to a point lies in the line where $$y$$ is zero. On the counter part, the y-intercept is a point in the line where $$x$$ is zero. However, in the second method, we simply transform the given equation into its equivalent slope - intercept form $$y=mx+b$$ where $$m$$ is the slope.