How to Determine the Slope of a Line

The slope--or slant--of a line can be quantified. You are probably used to referring to inclines as “steep” or “gradual,” but it is possible to give a numerical value to the degree of “steepness” or “gradualness” of a line. Only the basic mathematical operations of subtraction and division are needed to calculate the slope of any line if you have two points on the line. Determining the slope when you are given the formula of a line is simplicity itself, because the slope is part of the formula--or easily derived from the formula.

Instructions

Determine the Slope from a Graph or Points on a Line

1. Choose any two points on the line. Designate these points as point 1 and point 2. Name the coordinates of the first point (x1, y1) and designate the ordered pair of the second point (x2, y2). It doesn’t matter which point you call point 1 or point 2.

2. Write the equation for the difference of the y coordinates: y2 - y1. This is the vertical distance between the two points; it is how far the line rises in going from point 1 to point 2.

3. Put down the equation for the difference in the x coordinates: x2 - x1. This represents the horizontal distance the line runs in going from point 1 to point 2.

4. Calculate the slope, the rise per run: y2 - y1 / x2 - x1. This is how far up or down (vertical distance) the line goes per unit of horizontal distance.

5. Label your answer "m." The formula for the slope is thus: m = y2 - y1 / x2 - x1.

Determine the Slope from the Formula of a Line

1. Pick out the slope immediately if given the equation in the form y = mx + b. Remember, m is the slope. So, if you are asked to find the slope of a line whose equation is y = 3x + 10 , the slope is simply 3, the coefficient of x. Similarly, the slope of y = -½ x -5, is -½.

2. Use ordinary algebraic manipulation to find the slope if given the formula for a line in the form of Ax + By = C. Remember, that in simplifying an equation, you need to respect the equal sign, and whatever you do to one side of the equation, you must also do to the other.

3. Convert the equation above into one in the form of y = mx + b. Here’s how:Ax + By = C[-Ax] + Ax +By = C + [-Ax]By = C -AxBy = -Ax + Cy = -A/B x + C/BThe slope, then is -A/B.For the line, 2x + 3y = 12, solving for y yields y = -2/3 x + 4. The slope, therefore, is -2/3.

Tips & Warnings

• In choosing your two points, it often simplifies things, if you use the x and y intercepts (the points where the line crosses the x and y axes).

• The slope of a horizontal line is zero. When the line is horizontal, the y value is constant, y2 = y1, so y2 - y1 = 0. Vertical lines have no slope.