All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as [latex]x=4[/latex]. Then we checked the solution by substituting back into the equation.

Here’s an example of a linear equation in one variable, and its one solution.

[latex]\begin{array}{c}3x+5=17\hfill \\ \\ 3x=12\hfill \\ x=4\hfill \end{array}[/latex]

But equations can have more than one variable. Equations with two variables can be written in the general form [latex]Ax+By=C[/latex]. An equation of this form is called a linear equation in two variables.

Notice that the word “line” is in linear.

Here is an example of a linear equation in two variables, [latex]x[/latex] and [latex]y\text{:}[/latex]

[latex]\color{red}{A}x+\color{blue}{B}y=\color{green}{C}[/latex]

[latex]x+\color{blue}{4}y=\color{green}{8}[/latex]

[latex]\color{red}{A=1},\color{blue}{B=4},\color{green}{C=8}[/latex]

Is [latex]y=-5x+1[/latex] a linear equation? It does not appear to be in the form [latex]Ax+By=C[/latex]. But we could rewrite it in this form.

By rewriting [latex]y=-5x+1[/latex] as [latex]5x+y=1[/latex], we can see that it is a linear equation in two variables because it can be written in the form [latex]Ax+By=C[/latex].

Linear equations in two variables have infinitely many solutions. For every number that is substituted for [latex]x[/latex], there is a corresponding [latex]y[/latex] value. This pair of values is a solution to the linear equation and is represented by the ordered pair [latex]\left(x,y\right)[/latex]. When we substitute these values of [latex]x[/latex] and [latex]y[/latex] into the equation, the result is a true statement because the value on the left side is equal to the value on the right side.

In the next video you will see more examples of how to determine whether an ordered pair is a solution to a linear equation.

Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the [latex]x\text{- and }y\text{-values}[/latex] of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[/latex] and then solve the equation for [latex]y[/latex]. Or, choose a value for [latex]y[/latex] and then solve for [latex]x[/latex].

We’ll start by looking at the solutions to the equation [latex]y=5x - 1[/latex] we found in the previous chapter. We can summarize this information in a table of solutions.

To find a third solution, we’ll let [latex]x=2[/latex] and solve for [latex]y[/latex].

The ordered pair is a solution to [latex]y=5x - 1[/latex]. We will add it to the table.

We can find more solutions to the equation by substituting any value of [latex]x[/latex] or any value of [latex]y[/latex] and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.

Find Solutions to Linear Equations in Two Variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either [latex]x[/latex] or [latex]y[/latex]. We could choose [latex]1,100,1,000[/latex], or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose [latex]0[/latex] as one of our values.

We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation [latex]3x+2y=6[/latex].

Let’s find some solutions to another equation now.

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.