Solving Systems of Equations by Graphing
In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy 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 unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution. In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
A General Note: Types of Linear Systems
There are three types of systems of linear equations in two variables, and three types of solutions.- An independent system has exactly one solution pair [latex]\left(x,y\right)[/latex]. The point where the two lines intersect is the only solution.
- An inconsistent system has no solution. Notice that the two lines are parallel and will never intersect.
- A dependent system has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.
How To: Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.
- Substitute the ordered pair into each equation in the system.
- Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.
Example 1: Determining Whether an Ordered Pair Is a Solution to a System of Equations
Determine whether the ordered pair [latex]\left(5,1\right)[/latex] is a solution to the given system of equations.Solution
Substitute the ordered pair [latex]\left(5,1\right)[/latex] into both equations.Analysis of the Solution
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.Try It 1
Determine whether the ordered pair [latex]\left(8,5\right)[/latex] is a solution to the following system.Solving Systems of Equations by Graphing
There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.Example 2: Solving a System of Equations in Two Variables by Graphing
Solve the following system of equations by graphing. Identify the type of system.Solution
Solve the first equation for [latex]y[/latex].Try It 2
Solve the following system of equations by graphing.Q& A
Can graphing be used if the system is inconsistent or dependent?
Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.Licenses & Attributions
CC licensed content, Specific attribution
- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.