Graphing Equations

One can graph a line if we know only its slope and one point on it. First, plot the point. Next, if the slope is a fraction, move to the right the number of spaces equal to the denominator, and move up (or down, if the slope is negative) the number of spaces equal to the numerator. Plot a point at the spot you end up. If the slope is not a fraction, move 1 space to the right and then move up or down the number of spaces equal to the slope. You can also move left the number of spaces in the denominator and down (or up if negative) the number of spaces in the numerator and plot a point at the spot you end up.

Connecting these points with a straight line and extending on both sides yields a line with the desired slope and containing the given point.

Example 1. Graph the line which passes through (2, 3) and has a slope of .

Example 2. Graph the line which passes through (1, 1) and has a slope of -4.

We can find the slope of any line represented by an equation. Here are the steps:

1. Using inverse operations, get the two variables on opposite sides.
2. Write a fraction with the coefficient of x in the numerator and the coefficient of y in the denominator (this is counterintuitive, since we have been dealing with y in the numerator and x in the denominator)
3. This fraction is the slope.

Example 3. Find the slope of the line given by the equation 5y = - 2x + 1

1. 5y = - 2x + 1 (no change)
2. = -
3. m = -

Example 4. Find the slope of the line given by the equation 2y - x + 15 = 0

1. 2y + 15 = x
2. 
3. m =

Example 5. Find the slope of the line given by the equation 3 = - y - 15x

1. y + 3 = - 15x
2. = - 15
3. m = - 15

We can see from the steps for finding slope and from these examples that the constant term does not affect the slope.

Once we know the slope of an equation, we can find a point that satisfies the equation by plugging in a value for x and solving. Then we can plot this point and graph the equation.