How To Find Slopes

The slope is an important trait of lines and linear inequalities. Finding the slope is rather simple, requiring only the basic operations of arithmetic: addition, subtraction, multiplication and division. You have two general methods of finding a line's slope: calculating it from two points on the line and detecting it in the equation of the line.

Though people think of lines as visual objects, lines stem from equations. The slope of a line is one of the line's most important aspects, as it represents both the steepness and direction of the line. The magnitude, or size, of the slope represents steepness; the larger the number, the steeper the slope. The magnitude literally means how many units the slope moves up or down for every one unit right. The sign, either positive or negative, represents whether the slope is slanting upward or downward, respectively. For example, a slope of -5 represents a downward movement of 5 for every 1 unit right.

You can find a line's slope through a calculation involving any two points from that line. You can write two points from the line as (x1, y1) and (x2, y2). You find the slope by dividing the difference between the y-values by the difference between the x-values. That is, the formula (y2 – y1) / (x2 – x1) gives the slope.

Sometimes the slope is immediately obvious from the equation of the line. A line's equation is often in the form y = mx + b, the slope-intercept form. In this equation, "m" is the slope. Thus, for the line y = -2x + 4, -2 is the slope. If your line is not in the form y = mx + b, you can use algebra to put it in that form.

You should practice finding slopes rather than just memorizing methods. Assume you have the points (-3, 1) and (0, 7) from a line and want to find the line's slope. The formula (y2 – y1) / (x2 – x1) yields the calculation (7 – 1) / [0 - (-3)], which simplifies to 6 / (-3), or -2. Thus, -2 is the slope for the line on which (-3, 1) and (0, 7) lie. If you have the equation for a graphed line, such as 4x + 2y = 6, you can rewrite it as y = mx + b with algebraic operations. For this example, subtract 4x from both sides and then divide by 2. The result is y = -2x + 3. The m-value representing the slope is always next to the x, so in this case, the slope is -2.