Systems of three equations with three variables are often called 3-by-3 systems. In general, to find a single solution to any system of equations, you need as many equations as you have variables.

Recall from Geometry that the graph of a linear equation in three variables is a plane. When you graph a system of three linear equations in three dimensions, the result is three planes that may or may not intersect. The solution to the system is the set of points where all three planes intersect. These systems may have one, infinitely many, or no solution.

Identifying the exact solution from a graph of a 3-by-3 system can be very difficult. However, you can use the methods of elimination and substitution to reduce a 3-by-3 system to a 2-by-2 system and then use the methods that you learned in Algebra 1.

Example 1: Solving a Linear System in Three Variables Use elimination to solve the system of equations. 5x – 2y – 3z = –7 1 2x – 3y + z = –16 2 3x + 4y – 2z = 7 3 Step 1 Eliminate one variable. In this system, z is a reasonable choice to eliminate first because the coefficient of z in the second equation is 1 and z is easy to eliminate from the other equations.

Use equations and to create a second equation in x and y. Example 1 Continued 5x – 2y – 3z = –7 5x – 2y – 3z = –7 1 Multiply equation - by 3, and add to equation . 1 2 3(2x –3y + z = –16) 2 6x – 9y + 3z = –48 11x – 11y = –55 4 Use equations and to create a second equation in x and y. 3 2 1 3x + 4y – 2z = 7 3x + 4y – 2z = 7 3 Multiply equation - by 2, and add to equation . 3 2 2(2x –3y + z = –16) 4x – 6y + 2z = –32 2 7x – 2y = –25 5

You now have a 2-by-2 system. 7x – 2y = –25 Example 1 Continued 11x – 11y = –55 4 You now have a 2-by-2 system. 7x – 2y = –25 5

You can eliminate y by using methods from Lesson 3-2. Example 1 Continued Step 2 Eliminate another variable. Then solve for the remaining variable. You can eliminate y by using methods from Lesson 3-2. Multiply equation - by –2, and equation - by 11 and add. 4 5 –2(11x – 11y = –55) –22x + 22y = 110 4 1 11(7x – 2y = –25) 77x – 22y = –275 5 55x = –165 1 x = –3 Solve for x.

Step 3 Use one of the equations in your 2-by-2 system to solve for y. Example 1 Continued Step 3 Use one of the equations in your 2-by-2 system to solve for y. 11x – 11y = –55 4 1 11(–3) – 11y = –55 Substitute –3 for x. 1 y = 2 Solve for y.

Substitute –3 for x and 2 for y. 2(–3) – 3(2) + z = –16 Example 1 Continued Step 4 Substitute for x and y in one of the original equations to solve for z. 2x – 3y + z = –16 2 Substitute –3 for x and 2 for y. 2(–3) – 3(2) + z = –16 1 z = –4 Solve for z. 1 The solution is (–3, 2, –4).