3.5 Solving systems of equations in 3 variables

Select any 2 of the 3 equations and try to eliminate one variable to get an equation with 2 variables Select a different pair of equations and try to eliminate the same variable as step 1 Combine the equations found from step 1 and step 2 then solve for systems of 2 variable Substitute the solutions from step 3 to any of the original equations to solve for the last variable

2x – y + z = 10 -6x + 3y – 3z = -30 4x + 2y – 3z = 10 x – 3y + 2z = 8 Step 1: Select equation 1 and 2, then multiply the first equation by 2 (2) 2x – y + z = 10 4x – 2y + 2z = 20 4x + 2y – 3z = 10 8x – z = 30 Step 2: Select eq.1 and 3 (-3) 2x – y + z = 10 -6x + 3y – 3z = -30 x - 3y + 2z = 8 -5x - z = -22 Step 3: combine 8x – z = 30 -5x – z = -22 should get x = 4, z = 2 Step 4: Substitute to 2x – y + z = 10 8 - y + 2 = 10 10 – y = 10 y = 0 Solution: (4, 0, 2)

Try: 2x + y - 3z = -4 4x - 2y + z = 9 3x + 5y - 2z = 5 Answer: (2,1,3)

Try: x + y + z = 57 -2x + y = 3 x - z = 6 Answer: (15, 33, 9)