three variable system of equations solver

We’ll use equation 2 (after decimals were removed):Substitute the values for x and zSimplifySubtract 3 from both sizesMultiply by (-1)The solution for the system of equations is the ordered triple: Ĭheck your answerSubstitute the values for x, y and z into the other two original equations (1 and 3)Equation 1:Substitute the values for x, y, zSimplifyThe resulting statement is true.Equation 3:Substitute the values for x, y, zSimplifyThe resulting statement is true. Use one of the original equations to solve for yNext we need to substitute the values of x and z into one of the original equations to solve for y.

Solve first for z, then for xAdd the new set of equations A and B to eliminate x, and then solve for z:Add the two equations to eliminate xDivide both sides by 102 and reduceNow substitute the value for of z into equation A or B to solve for x We are using equation B:Substitute 1/6 for zSimplifySubtract 2 from both sidesDivide both sides by 10 and reduce However, we need one of those coefficients to be negative, so we’ll multiply the second equation by -4. If we multiply equation A by 5 and equation B by 4, the resulting coefficient of x will be 40. The least common multiple of 8 and 10 is 40, so we will want the coefficients of the x terms to be 40 and -40. It will need to be a common multiple of the current coefficients, 8 and 10. To do so we need to multiply each of the equations by some factor.Before we determine that factor, we need to determine what the resulting coefficient of x will be. Solve the system in two variablesTo solve our new system of equations A & B, the first step is to eliminate one of the variables If we choose to eliminate x by addition, we must get the equations in a form such that the coefficients of x in the two equations are inverses of one another (for example: +1 and -1 or +5 and -5). In the first column, we eliminated y from equations 1 & 2, resulting in equation A.In the second column, we eliminated y from equations 2 & 3, resulting in equation B. Then eliminate the same variable from another set of two equations. To do so, we eliminate one of the variables from two of the equations. This becomes our new #2Įliminate One VariableWe have rewritten our three equations as follows:Now, to solve, we need to first get two equations with two variables. Remove Fractions and DecimalsWe can eliminate the fraction in #1 by multiplying both sides of the equation by 4.This is our new version of #1We can eliminate the decimals in #2 by multiplying both sides of the equation by 100.Note that all terms have a common factor of 5 So, Let’s divide both sides by 5, so we have smaller numbers to work with. 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. Solving Systems of Linear equations with 3 VariablesTo solve for three variables, we need a system of three independent equations.

Three variable system of equations solver