How to Do 7th Grade Math: Finding Slopes

How to Find Slope

The slope, or steepness, of a line represents the rate at which the line is increasing or decreasing. A line with a slope that increases from left to right has a positive slope, and a line that decreases from left to right has a negative slope. Lines that are very steep have larger slopes compared to lines that are less steep.

The Point-Slope Formula

Sometimes, you'll be asked to find the slope of a line that's drawn on a graph or described in a problem. Usually, two points on the line will be given to you in the problem. For instance, your teacher might ask you to find the slope of a line that passes through (1, 2) and (4, 8).

The formula for slope is (change in y)/(change in x). It is often called the point-slope formula, and sometimes it's described as 'rise over run.' Rise refers to the change in the value of y between one point and the other point, and run is the change in the value of x between the two points. Here's another way of writing the point-slope formula for a line that passes through points (a, b) and (c, d):

Slope = (d - b)/(c - a)

Let's solve the problem stated above with the points (1, 2) and (4, 8). First, we need to find the 'change in y,' which is just the difference in the y values of the two points. The y values are eight and two, and 8 - 2 = 6, so the 'rise' in our 'rise over run' formula is six.

The 'run' is the 'change in x,' which is the difference between four and one. Since 4 - 1 = 3, three is the 'run.' Last, we divide 'rise over run' to get the slope of two (6 ÷ 3 = 2). Here's the mathematical solution:

(8 - 2)/(4 - 1)

= 6/3

= 2

Tip: The slope of a line that passes through (1, 2) and (4, 8) is the same as the slope of a line that passes through (4, 8) and (1, 2). It doesn't matter which order the points are in, but it does matter that you keep the points in the same order when calculating the values of the rise and run. The two values for each point should be lined up vertically in your equation. For instance, eight should be above four and two should be above one.

Slope-intercept Form

Every line is represented by a linear equation. You might be given an equation that represents a line and asked to find the slope. To do this, you don't have to actually draw the line and find two points. Instead, you just have to manipulate the equation so that it is in slope-intercept form. It looks like this:

y = mx + b

When an equation is in this form, the value of m is the slope. For instance, find the slope of a line with the equation -4x + y = -2. To put this equation in slope-intercept form, move the term '-4x' to the right side by adding it to both sides of the equation. This gives you the equation 'y = 4x - 2,' which is now in slope-intercept form. The value of m in this equation is four, so that is the slope.

Tip: Slopes come in many forms. They may be fractions or integers, and they can also be positive or negative. If an equation in slope-intercept form is y = x + b, then the slope, or value of m, is equal to one. For instance, the slope of y = x + 3 is one. The one is left out since (1)(x) = x.

Slopes for Vertical and Horizontal Lines

Since horizontal lines have no steepness, their slopes are equal to zero. Imagine a line that passes through the points (2, 3) and (4, 3). The 'change in y' is zero, since 3 - 3 = 0. Zero divided by any other number equals zero, so that is the slope.

For vertical lines, the slope is undefined. For instance, a line that passes through the points (5, 7) and (5, 3) would have a 'change in x' of zero. Since you can't divide by zero, the slope is undefined.

Practice Problems

1. Find the slope of a line that passes through the points (2, 1) and (6, 3).

2. Find the slope of a line with the equation y = 1/2x + 3.

3. Find the slope of a line that passes through the points (4, -2) and (6, -2).

4. Find the slope of a line with the equation 5x + y = 7.

Solutions

1. The correct answer is 1/2. Plug the numbers into the point-slope formula, like this:

(3 - 1)/(6 - 2)

= 2/4

= 1/2

2. This equation is already in slope-intercept form, so no manipulation is required. The fraction 1/2 is in the m position, so it is the slope.

3. The 'change in y' for this line is zero, since both y-values are the same (-2 - [-2] = 0). This means that the slope of the line is zero.

4. To find the slope of this line, we need to put the equation 5x + y = 7 into slope-intercept form. Here's how:

5x + y = 7

5x - 5x + y = - 5x + 7

y = - 5x + 7

Once the equation is in slope-intercept form, you can see that the value of m is -5, so that is the slope.