![]() One such method is solving a system of equations by the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. We will consider two more methods of solving a system of linear equations that are more precise than graphing. Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. Solving Systems of Equations by Substitution If the two lines are identical, the system has infinite solutions and is a dependent system. If the two lines are parallel, the system has no solution and is inconsistent. Yes, in both cases we can still graph the system to determine the type of system and solution. ![]() There are no points common to both lines hence, there is no solution to the system.Ĭan graphing be used if the system is inconsistent or dependent? 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.Ģ ( 4 ) + ( 7 ) = 15 True 3 ( 4 ) − ( 7 ) = 5 True 2 ( 4 ) + ( 7 ) = 15 True 3 ( 4 ) − ( 7 ) = 5 True 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. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. A skateboard manufacturer introduces a new line of boards.
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