# System of Linear Equations: Definition & Examples

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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

A system of linear equations can come in handy when we come across problems where we have more than one quantity to find. In this lesson, we'll learn how to define a system of linear equations and look at examples.

## Linear Equations

Imagine telling a friend about your favorite coffee and doughnut shop. Your friend asks how much it charges for a cup of coffee and a doughnut. While you can't remember how much each one costs individually, you do remember that the last time you ordered one coffee and two doughnuts, your bill was \$7.00. And today, when you bought two coffees and three doughnuts, your bill came to \$12.00.

Believe it or not, this is all the information you need to answer your friend's question. To do this, we use what is called a system of linear equations. A linear equation is a polynomial equation in which the unknown variables have a degree of one. That is, all of the unknown variables in a linear equation are raised to the power of one. Some examples of a linear equation are shown in the image below.

These equations are polynomial equations in which the variables are raised to the power of one. A couple of non-examples are shown in this other image below.

In the equation, 2x^2 + 3y - 4 = 0, the variable x is raised to the power of 2, so this is not a linear equation. The equation, y = 3x / (x - 1), isn't a linear equation because, while its variables are raised to the power of one, it isn't a polynomial equation.

## System of Linear Equations

A system of linear equations is a set of two or more linear equations with the same variables. For example, the sets in the image below are systems of linear equations.

Let's return to the question your friend asked about the cost of a cup of coffee and a doughnut at your favorite coffee shop. Here, c represents the cost of one coffee, and d represents the cost of one doughnut. The linear equation for your purchase of one coffee and two doughnuts, which came to \$7.00, can be written as follows: c + 2d = 7. Similarly, your purchase of two coffees and three doughnuts, for a total of \$12.00, can be expressed as: 2c + 3d = 12.

Now, let's put these two equations together. In partnership, they give us the system of linear equations required to figure out the cost of one coffee and the cost of one doughnut.

2c + 3d = 12

c + 2d = 7

## System of Linear Equations Solution

A solution to a system of linear equations is a set of numbers that, when we substitute numbers for specified variables in the system, makes each equation in the system a true statement. For example, if we plug 4 in for x and 7 in for y, both of the equations in the following system are true statements.

x - y = -3

x + y = 11

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