Linear Functions
Input, Output and Graphs
Slope and Intercept
Parallel and Perpendicular Lines
Systems of Equations

Share

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

Linear FunctionsSlope and Intercept

Reading time: ~15 min

Common Core Standard 8.EE.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.

Here you can see a coordinate system, with a straight line that goes through its origin. To get started, pick a point anywhere on the line.

We can draw a right-angled triangle between this point and the origin of the coordinate system.

Try sliding the point along the line: notice how different points form differently sized triangles, but they are all . The best way to see this is to look at the two angles along the x-axis. They are always the same size, so by the AA-condition the triangles must all be similar.

Now we can use one of the results we know about similar triangles: the ratio of two of the sides is always constant. Move the point again, and watch what happens:

${p.y}${p.x}=${p.y/p.x || '???'}

But the opposite is also true: any point (x, y) that satisfies this equation must lie somewhere on the line. Therefore we now have an “equation” for the line:

yx=1.5

⇔y=1.5x

It turns out that every line that goes through the origin of a coordinate system has an equation of the form y=mx, where m is called the slope.

If you’re given a line, you can find the corresponding value of m by picking any point that lies on the line and simply dividing its y and x value. Here are a few examples:

m =

m =

m =

But what about lines that don’t go through the origin of the coordinate system? In that case we need one more component: we can take the line with the same slope that goes through the origin, and shift it along the y-axis by adding or subtracting a number:

y=23x ${sign(a)} ${abs(a)}

As you can see above, the number added to the value of y is the same as the distance between the origin of the coordinate system, and the point where the line crosses the .

We now have an equation for any (non-vertical) line in the coordinate plane:

y=mx+b,

where m and b are two numbers we have to fill in. As you saw before, m is the slope of the line, and b is the y-axis intercept.

If you’re given any line, like the one on the left, you can find the value of b by looking at the point where the line crosses the y-axis. In this example, b = .

Similarly, you can find the slope m by drawing any triangle below the line, and dividing its height and base. In this example, the slope is m = .

In other words, the equation of this line is

y=

Here are a few more exercises. Can you find the slope and y-intercept in each case, and write down the equation of the line?

y=

y=

y=