Solve the system.EXAMPLESolve the system: Strategy Since the coefficients of the z-terms a...

Solve the system.

EXAMPLE

Solve the system:

Strategy Since the coefficients of the z-terms are opposites in the second and third equations, we will add the left and right sides of those equations to eliminate z. Then we will choose another pair of equations and eliminate z again.

WHY The result will be a system of two equations in x and y that we can solve by addition.

Solution

Step 1 We can skip step 1 because each equation is written in standard form and there are no fractions or decimals to clear. We will number each equation and move to step 2.

Step 2 If we pick equations 2 and 3 and add them, the variable z is eliminated.

Step 3 We now pick a different pair of equations (equations 1 and 3) and eliminate z again. If each side of equation 3 is multiplied by 2, and the resulting equation is added to equation 1, z is eliminated.

Step 4 Equations 4 and 5 form a system of two equations in two variables, x and y.

To solve this system, we multiply equation 4 by −5 and add the resulting equation to equation 5 to eliminate y:

To find y, we substitute 3 for x in any equation containing x and y only (such as equation 5) and solve for y:

Step 5 To find z, we substitute 3 for x and 2 for y in any equation containing x, y, and z (such as equation 1) and solve for z:

The solution of the system is (x, y, z) = (3, 2, 1). Because this system has a solution, it is a consistent system.

Step 6 Verify that these values satisfy each equation in the original system.