880 CHAPTER 9 Systems and Matrices Solving a Linear System with Three Unknowns Step 1 Eliminate a variable from any two of the equations. Step 2 Eliminate the same variable from a different pair of equations. Step 3 Eliminate a second variable using the resulting two equations in two variables to obtain an equation with just one variable whose value we can now determine. Step 4 Find the values of the remaining variables by substitution. Write the solution of the system as an ordered triple. EXAMPLE 6 Solving a System of Three Equations with Three Variables Solve the system. 3x + 9y + 6z = 3 (1) 2x + y - z = 2 (2) x + y + z = 2 (3) SOLUTION Step 1 Eliminate z by adding equations (2) and (3). 3x + 2y = 4 (4) Step 2 To eliminate z from another pair of equations, multiply each side of equation (2) by 6 and add the result to equation (1). 12x + 6y - 6z = 12 Multiply (2) by 6. 3x + 9y + 6z = 3 (1) 15x + 15y = 15 (5) Step 3 To eliminate x from equations (4) and (5), multiply each side of equation (4) by -5 and add the result to equation (5). Solve the resulting equation for y. -15x - 10y = -20 Multiply (4) by -5. 15x + 15y = 15 (5) 5y = -5 Add. y = -1 Divide by 5. Step 4 Let y = -1 to find x using equation (4) by substitution. 3x + 2y = 4 (4) 3x + 21-12 = 4 Let y = -1. x = 2 Solve for x. Now substitute 2 for x and -1 for y in equation (3) to find z. x + y + z = 2 (3) 2 + 1-12 + z = 2 Let x = 2, y = -1. z = 1 Solve for z. Verify that the ordered triple 12, -1, 12 satisfies all three equations in the original system. The solution set is 512, -1, 126. This is a consistent system with a single solution. See Figure 6(a). S Now Try Exercise 47. Make sure equation (5) has the same two variables as equation (4). A single solution P I II III Figure 6(a) (repeated) Write the values of x, y, and z in the correct order.
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