![]() If we eliminate the fractions and decimals, the equations will be easier to work with. Put equations in Standard FormRewriting Equation 3 into standard form:Add y to both sizesAdd 6z to both sidesOur 3 equations are now:But, note that equations #1 contains a fraction, and #2 contains decimals. We need to get the y and z terms to the left side of the equation. Note that equation #3 is not in standard form. all variables are on the left side of the equation. Result will be a second equation in two variables.Solve the new system of two equations.Using the solution for the two variables, substitute the values into one of the original equations to solve for the third variable.Check the solution set in the remaining two original equations.Įxample system of three equationsExample: solve for x, y and zFirst, we need to ensure that all equations are in standard form, i.e. Result will be a new equation with two variables.Eliminate the same variable using another set of two equations. Steps to solving a system of equations in 3 VariablesEnsure that the equations are in standard form: Ax + By + Cz = DRemove any decimals or fractions from the equations.Eliminate one of the variables using two of the three equations. Independent, Dependent and Inconsistent EquationsSystems of three equations in three variables follow the same characteristics of systems of equations in two variablesIndependent equations have one solution Dependent equations have an infinite number of solutionsInconsistent equations have no solutionSolving a system of equations in three variables involves a few more steps, but is essentially the same process as for systems of two equations in two variables. If we multiply both sides of the first equation by -3, then we will be able to eliminate the x -variable.Solving Systems of Linear equations with 3 VariablesTo solve for three variables, we need a system of three independent equations. In this case, let’s focus on eliminating x. With the addition method, we want to eliminate one of the variables by adding the equations. The graphs of the equations in this example are shown below. Parallel lines will never intersect thus, the two lines have no points in common. Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. 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. (credit: Thomas Sørenes) Introduction to Solutions of Systems
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