Systems of Equations

A system of linear equations consists of two or more linear equations in two or more variables such that all equations in the system are considered simultaneously. To find the single, unique solution of a system of linear equations, we must find a numerical value for each variable in the system that satisfies all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a single, unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a single, unique solution.

In this section, we will look at systems of linear equations in two variables, which consist of two equations that each contain the same two variables. For example, consider this system of linear equations in two variables:

The solution of a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair [latex](4, 7)[/latex] is the solution of the system of linear equations. We can verify the solution by substituting the value of each variable into each equation to see if the ordered pair satisfies both equations:

This system consists of two linear equations, so graphically, we are considering two lines. Each point on the first line is a solution of the first equation. Each point on the second line is a solution of the second equation. Only a point on both lines will satisfy both equations. For the system we just looked at, the two lines cross at the solution (4, 7), as shown in figure 1.

Figure 1. The intersection point of the two lines is the solution of the system.

How To determine whether an ordered pair is a solution of a system

1. Substitute the ordered pair into each equation in the system.
2. Determine whether true statements result from the substitution in each equation; if so, the ordered pair is a solution.

Example 1

Determine whether the ordered pair [latex]\left(5,1\right)[/latex] is a solution of the system of equations:

[latex]\begin{array}{l}x+3y=8\hfill \\ 2x - 9=y\hfill \end{array}[/latex]

Solution

Substitute the ordered pair [latex]\left(5,1\right)[/latex] into each equation:

[latex]\begin{array}{ll}\left(5\right)+3\left(1\right)=8\hfill & \hfill \\ \text{ }8=8\hfill & \text{True}\hfill \\2\left(5\right)-9=\left(1\right)\hfill & \hfill \\ \text{ }\text{1=1}\hfill & \text{True}\hfill \end{array}[/latex]

The ordered pair [latex]\left(5,1\right)[/latex] satisfies both equations, so it is the solution of the system.

We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.

The following video shows another example of how to verify whether an ordered pair is a solution of a system of equations.

Classifying a System of Linear Equations in 2 Variables

In addition to considering the number of equations and variables, we can categorize a system of linear equations by the number of solutions it has. A system of equations is said to be consistent if it has at least one solution.

A consistent system is considered to be independent if it has a single solution, like the example we just explored. Graphically, this means that the two lines that make up the system have different slopes and intersect at exactly one point in the plane (e.g., figure 1).

A consistent system is considered to be dependent if the two lines that make up the system have the same slope and the same [latex]y[/latex]-intercept. In other words, the lines coincide, so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions of the system that all lie on the same line.

A system of linear equations is said to be inconsistent if it has no solution. This is where the graph of the system consists of two parallel lines. The lines have the same slope and different [latex]y[/latex]–intercepts. There are no points common to both lines; hence, there is no solution to the system.