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Basic Algebra

Basic Algebra

At a Glance - Graphing Lines

You've now worked pretty extensively with equations containing one variable. We are now going to briefly work with ones involving two variables, x and y.

Equations that have an x term and a y term, or only one of the two, can be graphed as lines on a basic coordinate grid.

You may see equations that look like this:

y = -2x + 3

or this:

2x + y = 3

Both are different representations of the same line; in the 2nd equation the -2x is moved to the other side. The first equation is in what mathematical people call "slope-intercept form" and the second is in "standard form" (more on these later).


Slope: (usually represented by the variable m) of a line measures its steepness; the larger the absolute value of the slope, the greater the steepness.

Slope is calculated as change in the vertical direction (y) ÷ change in the horizontal (x); this is often called rise over run.

m = (change in y)/(change in x) = rise/run

To find the slope you can pick two points on the line and find the difference in the y values, and divide it by the difference in the x values. The important thing to remember is to keep the points in the same order in the numerator and denominator.


A positive slope tells us that the line goes uphill, from low to high. A negative slope tells us that the line goes downhill, from high to low.

different slopes

The two graphs on the left have steeper slopes, so the absolute value of their slopes is larger than the absolute value of the slopes on the right.

To calculate the slope of a line, start by picking any two points on the line. Calculate the vertical difference and divide by the horizontal difference, you can do this by counting, or by using the formula. If you use the counting method, don't forget to include the sign, positive if it slopes uphill and negative if it slopes downhill (the formula will do this for you).

Example 1

Find the slope of this line:

slope 1

Example 2

Find the slope of this line:

slope 2

Example 3

Find the slope of this line:

slope 3

Example 4

Find the slope of this line:

slope 4

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