Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit.The solution to a system of dependent equations will always be true because both equations describe the same line.Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect.It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together.A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables.In this method, we solve for one variable in one equation and substitute the result into the second equation. Another method of solving a system of linear equations is by substitution.In this method, we graph the equations on the same set of axes. One method of solving a system of linear equations in two variables is by graphing.Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.There are no points common to both lines hence, there is no solution to the system. The lines have the same slope and different y-intercepts. Thus, there are an infinite number of solutions.Īnother type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. Every point on the line represents a coordinate pair that satisfies the system. In other words, the lines coincide so the equations represent the same line. A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. A consistent system of equations has at least one solution.
In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. For example, consider the following system of linear equations in two variables. In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. Even so, this does not guarantee a unique solution. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Some linear systems may not have a solution and others may have an infinite number of solutions. 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. 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. 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.