Presentation on theme: "Solving Systems of three variables"— Presentation transcript:
1Solving Systems of three variables Use the answers in the original equations to solve the remaining variables.Solve the new system of equations by elimination or substitution.Eliminate one of the variables using 2 different equations 2 times
2Even though there is not a set order or rules to these problems I will try an give you some guidelines.1.) You want to get three equations with 3 variables down to 2 equations with 2 variables.a) on the easier problems you might be able to eliminate one variable from 2 of the 3 equations and the third equation might already be missing the same variable.b) The hardest of problems will require you to eliminate one variable from any 2 of the 3 equations then using elimination again eliminate that same variable using the equation you did not use in the first time with one of the equations you did use the first time.2.) Now you have 2 equations with 2 variables so use elimination or substitution to solve for one variable3.) Once you have solved for one variable substitute that value into one of the 2 equations with 2 variables and solve for the 2nd variable.4.) Lastly use the 2 values you have found in one of the original 3 equations and solve for the last variable.
3x + 2y – 3z = 152x – 2z = 6x + z = 3Now use x = 3 to substitute in to solve for the z.x 22x – 2z = 62x + 2z = 63 + z = 3(3,6,0)4x = 12z = 0Now use x = 3 and z = 0 to solve for y.x = 33 + 2y -3(0)= 153 +2y + 0 = 152y = 12y = 6
4I know it can be long but its nothing to bang your head over.
5Use the elimination method EXAMPLE 1Use the elimination methodSolve the system.4x + 2y + 3z = 1Equation 12x – 3y + 5z = –14Equation 26x – y + 4z = –1Equation 3SOLUTIONSTEP 1Rewrite the system as a linear system in twovariables.4x + 2y + 3z = 112x – 2y + 8z = –2Add 2 times Equation 3to Equation 1.16x z = –1New Equation 1
6Use the elimination method EXAMPLE 1Use the elimination method2x – 3y + 5z = –14Add – 3 times Equation 3to Equation 2.–18x + 3y –12z = 3–16x – 7z = –11New Equation 2STEP 2Solve the new linear system for both of its variables.16x + 11z = –1Add new Equation 1and new Equation 2.–16x – 7z = –114z = –12z = –3Solve for z.x = 2Substitute into newEquation 1 or 2 to find x.
7Use the elimination method EXAMPLE 1Use the elimination methodSTEP 3Substitute x = 2 and z = – 3 into an originalequation and solve for y.6x – y + 4z = –1Write original Equation 3.6(2) – y + 4(–3) = –1Substitute 2 for x and –3for z.y = 1Solve for y.Solution ( 2,-3,1)
83x + y +z = 14 -x + 2y – 3z = -9 5x - y + 5z = 30