3 by 3 Linear Systems
Geometry of 3X3 systems Recall that for lines, either they intersect in a point, are parallel, or are the same line. Similarly, if we have three planes either they intersect in a point, a line, don't intersect at all, or are the same planes. Therefore when we solve three linear equations and three unknowns, we can expect that the solution is a point, a line, no solution, or a plane.
Step by step rules for the elimination method for solving a three by three linear system Example: x + 2y - z = 2 Equation 1 2x + 2y + 2z = 12 Equation2 x - y + 2z = 5 Equation 3
Step 1: Choose the most convenient variable to eliminate In our example any seem convenient.
Step 2: Use any two of the equation to eliminate the variable. Then use a different pair to eliminate the same variable. The result is a 2 by 2 system. Multiply equation 1 by two and subtract from equation 2: 2x + 2y + 2z = 12
2x + 4y - 2z
= 4 -2y + 4z = 8 and Multiply equation 3 by 2 and subtract from equation 2 2x + 2y + 2z = 12
2x - 2y + 4z
= 10 4y - 2z = 2
Step 3: Use elimination to solve the system of the two equations that you found. 2y + 4z = 8 4y - 2z = 2 We multiply the second equation by 2 and add it to the first equation: -2y + 4z = 8
8y - 4z
= 4 6y = 12 y = 2 Re-substituting: -2(2) + 4z = 8 so that 4z = 12, z = 3
Step 4: Substitute the two values into any of the three equations the get the third value. x + 2(2) - (3) = 2 so x = 1
Step 5: Substitute all three values into the three equations to check your work.
Step 6: Reread the question and answer it.
Exercise Solve 3x -2y + 5z = 2 4x - 7y - z = 19 5x - 6y + 4z = 13
Application
Chris invests $2,200 into three accounts that pay 6%, 8% and 9% in annual interest. He has three times as much invested at 9% as he does at 6%. If his total interest for the year is $178, how much is invested at each rate?
Solution: Let x = the amount in the 6% account y = the amount in the 8% account z = the amount in the 9% account then x + y + z = 2,200 3x - z = 0 .06x + .08y + .09z = 178
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