Question

How do you find slope?

Answer 1

The slope of a line is a value that indicates both the direction and the steepness of the line. In a linear function graph in a cartesian grid the slope tells us how much does the value of y increases or decreases when the value of x increases in one unit. The slope appears in the equation of a line when this is written in the slope-intercept form:

`y=mx+b`

And in the point-slope form:

`y-k=m\cdot(x-h)`

In both cases the letter m expresses the slope, b is the y-intercept and (h,k) is a point in the line. Any point (x,y) that is part of the line is a solution to this equations. When you are given a table of values or the graph of the line and you need to find the slope you just need to take two points and use them to build two equation (you can use the point-slope or the slope-intercept form). For example, let's assume that you are given the graph of a line that passes through (1,2) and (3,8). We can take the general equation in slope-intercept form and build two equations replacing x and y:

`\begin{gathered} 2=m\cdot1+b \\ 8=m\cdot3+b \end{gathered}`

By solving this system of equations you can find the slope. For example we can substract m from the first equation:

`\begin{gathered} 2=m+b \\ 2-m=m-m+b \\ 2-m=b \end{gathered}`

And replace this in place of b in the second equation:

`\begin{gathered} 8=3m+b \\ 8=3m+2-m \\ 8=2m+2 \end{gathered}`

We substract 2 from both sides and then we divide both side by 2:

`\begin{gathered} 8=2m+2 \\ 8-2=2m+2-2 \\ 6=2m \\ (6)/(2)=(2m)/(2) \\ 3=m \end{gathered}`

Then for this case the slope is 3. You can do something similar using the equation in point-slope form remembering that (h,k) is a point of the line. We use one of the two points we have, for example (1,2) and we get:

`\begin{gathered} y-k=m\cdot(x-h) \\ y-2=m\cdot(x-1) \end{gathered}`

Then we use the other point taking (x,y)=(3,8) and we have an equation for m:

`\begin{gathered} (x,y)=(3,8) \\ 8-2=m\cdot(3-1) \\ 6=2m \end{gathered}`

If we divide by two we get:

`\begin{gathered} (6)/(2)=(2m)/(2) \\ 3=m \end{gathered}`

And we arrived at the same result. That's how you can find the slope of a line using two of its points. These points can be given by the question, found in a table of values or in a graph.