Inconsistent equations : two or more equations are inconsistent when they do not share any solutions, meaning their lines have no points in common. For example: and are consistent because they share one solution. This occurs when all the equations in the system are the same or are variations of the same equation and, therefore, represent the same line.Ĭonsistent equations : two or more equations are consistent when they share one or infinite solutions. Infinite solutions : There are an infinite number of variables that would make all the equations in the systems true. The point where they cross is the solution to the system. On a graph, the lines representing the equations cross once. One solution : There is one set of variables that would make all the equations in the system true. If they are linear equations, these lines would run parallel to each other. On a graph, the lines representing the equations do not touch. No solution : There are no variables that would make all the equations in the system true. There are three possible solution types for systems of linear equations: Repeat as necessary, such as when there are more than two linear equations in the system. Plug the resulting variable into either of the original equations and solve:Ĥ. Plug the resulting variable into the other equation and solve:ģ. Solve for or in one of the equations by isolating the variable:Ģ. Main steps for solving a system of linear equations by substitution:ġ. The variables that satisfy both equations are and orĦ.
Plug this variable into one of the original equations and simplify to isolate the remaining variable: Solve the equation to isolate the remaining variable:ĥ. Add or subtract the equations to eliminate their common variable:Ĥ. Multiply one or both of the equations by non-zero numbers that would make one set of terms cancel each other out if added or subtracted:ģ. Rewrite the equations so the variables are in the same order:Ģ. Main steps for solving a system of linear equations by elimination:ġ. These are called systems of linear equations and we can find their variable(s) using one of two methods: elimination and substitution. Sometimes we are given two or more equations that can be made true by the same variable or variables. If you have access to a graphing calculator, try graphing each equation and comparing the results. Though these equations may all look different, they all actually represent the same line. Important: In this form, and cannot both be zero ( ). It usually has constants and variables, which cannot contain exponents or roots, and is usually written in one of the following ways: A linear equation is an equation that represents a straight line.
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