The Graph of a Three Variable Equation

1. The answer is a plane.   Below is part of the graph of x + y + z = 3. The three solution points above are labeled: When we graph such an equation, we need three axes instead of two: the x-axis, the y-axis , and the z-axis. By convention, we let the z-axis be vertical and the y axis be horizontal. The x-axis  is perpendicular to these two. (like the corner of a room). Don’t worry, we will not be doing any graphing. The Graph of a Three Variable Equation Recall That a solution to an equation with two variables is an ordered pair.   For example, one solution to the equation 2x + 3y = 5 would be (x,y) = (1,1). Similarly, a solution to an equation with three variables, such as 3x – 2y + z = 7,  would be an ordered triple(x,y,z). Note that three such solutions to this equation would be the ordered triples: (2,1,3),  (1,-2,0) , (0,1,9). We know that when we graph all of the solutions to a two variable equation, we get a line. What do we get when we graph all of the solutions to a three variable equation?

2. Systems of Three Variable Equations Recall that when we try to solve a system of two variable equations, our solution may be the point the two lines have in common.  Sometimes the lines are parallel (there is no solution). Sometimes the equations in the system represent the same line (infinite solutions). When we solve a system of three variable equations, we can get intersecting or parallel planes. The following illustrates the variety of geometric configurations.

3. Step 5. Check the solution in all three of the given equations. Next Slide Procedure: Solving a System of Three Linear Equations in Three Variables Step 1. Write the three equations in the form Ax + By + Cz =D. Step 2. Choose two of the equations. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either x, y or z is zero. (Eliminate one of the variables) Step 3. Use the third equation that was not used in step 2 and either one of the two equations that were used from step 2. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of the same variable from step 2 is zero. (Eliminate the same variable) Step 4. We now have two equations with the two variables. Solve the system of two equations as was performed in the previous section.

4. Example 1. Solve: Answer: {(1, 2, 3)} Check answer in all three equations. Your Turn Problem #1 Solve the system Our goal is to obtain two equations with the same two variables. This example is already given in that form. Use the two equations with the two variables an solve for y. (eliminate z) Next substitute y=2 into either of the two equations with two variables to find z. Then substitute y=2 and z= 3 into row 1 to solve for z.

5. Decide on which variable to eliminate. We will get rid of z first in this example. Add row 1 and row 2 together. Example 2. Solve: * * Answer: {(2, -1, 3)} Check answer in all three equations. Now we need to use row 3 and either of the first two rows to get rid of z again (use row 1). Multiply row 1 by 2 and add it to row 3. Next, use the two new equations with two variables and solve as in the previous section. By adding the two equations together, the y will cancel and we obtain x. Next substitute x=2 into either of the two equations with two variables to find y. Then substitute x=2 and y= -1 into one of the original equations to solve for z. Use Row 1.

6. Example 3. Solve: * * ) ( Your Turn Problem #2 Solve the system Decide on which variable to eliminate. We will get rid of z first in this example. Add row 1 and row 3 together. Now we need to use row 2 and either row 1 or row 2 to get rid of z again (use row 3). Multiply row 3 by 2 and add it to row 2. Next, use the two new equations with two variables and solve as in the previous section. Next, substitute y=2 into either of the two equations with two variables to find x. Then substitute x=3 and y= 2 into one of the original equations to solve for z. Use Row 1. Answer: {(3, 2, 4)} The End. B.R. 1-15-07