3Here is a system of three linear equations in three variables: The ordered triple (2,-1,1) is a solution to this system since it is a solution to all three equations.
4The graph of a linear equation in three variables is a plane The graph of a linear equation in three variables is a plane. Three planes in space can intersect in different ways (pg 152).The planes could intersect in a line. The system has infinitely many solutionsThe planes could intersect in a single point. The system has exactly one solutionThe planes could have NO point of intersection. The left figure shows planes that intersect pairwise, but all 3 do not have a common point of intersection. The right figure shows parallel planes. Each system has NO solution.
5The linear combination method in lesson 3 The linear combination method in lesson 3.2 can be extended to solve a system of linear equations in three variables.
6Solve this systemOur strategy will be to use two of the equations to eliminate one of the variables.We will then use two other equations to eliminate the same variable.Once we have two equations with two variables, we can use the technique we learned in lesson 3.2
7Solve this systemOur strategy will be to use two of the equations to eliminate one of the variables.We will then use two other equations to eliminate the same variable.Once we have two equations with two variables, we can use the technique we learned in lesson 3.2
8Solve this system 7x +10z = 19 -x -4z = -13 2 -18z=-72 or z = 4 Equation 1Equation 22-3Equation 3Multiply Eq. 2 by 2 and add it to Eq. 1. Save this resultSolve this new system of linear equation in two variables.Multiply the bottom eq. by 7 and add it to the top eq.7x z = 19Now multiply Eq. 2 by -3 and add it to Eq. 3. Save this result.-18z=-72 or z = 4Substituting z=4 into either of the new equations will give x = -3……finally substituting these values into any of the original equations give y = 2.-x z = -13Our final solution is (-3,2,4)
9Here is a system with No Solution Here is a system with No Solution. Watch what happens when we try to solve it.Equation 1Equation 2Equation 3Add -3 times Eq 1 to Eq 2.Since this is a false equation, you can conclude the original system of equations has no solution.0=8
10Here is a system with MANY solutions Here is a system with MANY solutions. Watch what happens when we try to solve it.Equation 1Equation 2Solving this new system of two equation by adding -3 times the first eq. to 2 times the second eq. produces the identity 0 = 0. So, the system has infinitely many solution.Equation 3Add Eq. 1 to Eq. 2You could describe the solution this way: divide New Eq 1 by 2 to get x+y=2, or y=-x+2. Substituting this into the orignial Equation 1 produces z = 0. So any ordered triple of the form (x, -x+2,0) is a solution. For example (0,2,0) and (2,0,0) are solutions.2x + 2y = 4New EQ. 1Add Eq 2 to Eq 33x +3y = 6New EQ 2
11Substitution MethodSince x+y=z, substitute this for z in the first two equationsSimplifyFinally, solve this linear system of two equations and two variables to get x = 4 and y =8Since z=x+y, z = 12. Our final solution is (4,8,12)