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3 by 3 Linear Systems I. Midterm I II. 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 number, a line of numbers, no solution, or all points. III. Elimination: Step by step rules for the elimination method for solving a three by three linear system Example: x + 2y - z = 2 2x + 2y + 2z = 12 x - y + 2z = 5 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 Resubstituting: -2(2) + 4z = 8 so that 4z = 12, z = 3 Step 4: Substitute the two values into any of the three equation 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. The class will try the example: 3x -2y + 5z = 2 4x - 7y - z = 19 5x - 6y + 4z = 13
IV) Application
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