1. Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables A ___________________________ x, y, and z is an equation of the form ax + by + cz = d where a, b, and c are not all zero. The following is an example of a ________________________ in 3 variables: 2x + 5y – z = -7 x – 3y + z = 10 9x + y – 4z = -1 The solution to a system of equation in 3 variables is called an ______________ (x, y, z) whose coordinates make all 3 equations true. The _______ of a linear system in 3 variables is a plane in three-dimensional space. The intersection of the 3 planes determines the number of solutions. The planes intersect in a ____________. The planes intersect in a ______ or are the _____________. (Infinitely Many Solutions) The planes have __________________ of intersection. ( No Solution) linear equation in 3 variables system of linear equations ordered triple graph single point linesame plane no common point

2. Steps: 1. Rewrite 2 of the equations in 3 variables as equations in 2 variables using substitution or elimination. 2. Solve those 2 equations for both variables like you were taught in section 3.2. 3. Once you get the values from step 2, substitute them in to one of the original equations and solve for the 3 rd value. Then write your ordered triple. ** If the variables disappear and you get a false statement -3 = 0, then the system has no solutions. ** If the variables disappear and you get a true statement 0 = 0, then the system has infinitely many ordered triple solutions.

3. Examples: 1. 2x – y + 6z = -4 6x + 4y – 5z = -7 -4x – 2y + 5z = 9 -y = -2x – 6z – 4 y = 2x + 6z + 4 6x + 4(2x + 6z + 4) – 5z = -7 6x + 8x + 24z + 16 – 5z = -7 14x + 19z = - 23 -4x – 2(2x + 6z + 4) + 5z = 9 -4x – 4x – 12z – 8 + 5z = 9 -8x – 7z = 17

4. 2. 3x + y – 2z = 10 6x – 2y + z = -2 x + 4y + 3z = 7

5. 3. x + y – z = 2 3x + 3y – 3z = 8 2x – y + 4z = 7

6. 4. x + y + z = 6 x – y + z = 6 4x + y + 4z = 24