Negative Coefficients: Definition & Examples

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will discuss coefficients and negative numbers. Then, we will define negative coefficients and go over specific example of when they are used in math.

Negativity

Have you ever met a negative person? Someone who just cannot say a nice thing. They are the opposite of a positive person who is always upbeat and kind. Just like a negative person, negative numbers are opposite of positive numbers.

Negative numbers are those less than zero while positive numbers are greater than zero. -5, -1,000, and -1/4 are all examples of negative numbers. On a number line, any negative number will be to the left of zero.

Number line with negative numbers in red
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Coefficients

Coefficients are numbers that are multiplied by variables. Variables are letters that represent unknown numbers in math. Together, coefficients and variables make up terms such as 3x or -5y. Coefficients can be fractions, whole numbers, positive numbers, negative numbers, imaginary numbers, and so on.

Coefficients, Variables, and Terms
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Negative coefficients are simply coefficients that are negative numbers. An example of a negative coefficient would be -8 in the term -8z or -11 in the term -11xy. The number being multiplied by the variables is negative.

Equations and Expressions

Let's look at some examples of negative coefficients that you might come across in math. Often, you will see expressions or equations with negative coefficients.

Example 1:

-6x + 9y = 10

In the equation above, you can see the negative coefficient of -6 in front of the variable x. The coefficient of 9 in front of y is a positive coefficient.

Example 2:

-4z + 12z - 3y - 6y

Sometimes, you might have to simplify problems with negative coefficients. You can simplify the expression above by combining like terms. Like terms are terms with the same variable and exponent. For example, -4z and 12z are like terms because they both have the same variable z with an exponent of one.

When a variable has an exponent of one, you don't usually write one. For example z1 is just written as z.

To simplify this expression, we can combine the like terms of -4z and 12z by adding them. To do so, we will add the coefficients of -4 and 12 to get 8. The z stays as is, and the combined terms will equal 8z.

Likewise, we can combine -3y and -6y. These two negative coefficients will combine to give -9y. Thus, our simplified expression will be -9y + 8z. In math, it is customary to put the variables in alphabetical order, which is why the y terms would come first.

Graphing

Negative coefficients can mean a lot when it comes to graphing a function. For linear functions, a negative coefficient in front of the x means the m value or slope is negative. A negative slope will change how the line is graphed.

For example, the function y = 3x + 2 will graph as a line that goes up and to the right, while the function y = -3x + 2 will graph as a line that goes down as it moves from left to right. This can be seen in the picture below. The first function with a positive coefficient before the x variable is graphed in blue. This function has a positive slope. The second function with a negative coefficient before the x can be seen in red. This function has a negative slope.


Linear graphs with positive and negative coefficients
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