pencil red pen highlighter notebook calculator Solve the
pencil, red pen, highlighter, notebook, calculator Solve the system of equations. 2 x – y + 3 z = – 4 x + 2 y – 5 z = 11 x + 3 y – 2 z = 5 A B C Pick any two equations to eliminate a variable. Let’s eliminate x. Choose B and C x + 2 y – 5 z = 11 B – 1(x x ++3 y 3 y–– 2 z 2 z) == 5 – 1(5) C Label the new equation D. x + 2 y – 5 z = 11 –x – 3 y + 2 z = – 5 –y +1 – 3 z = 6 D +1 total:
2 x – y + 3 z = – 4 x + 2 y – 5 z = 11 x + 3 y – 2 z = 5 A B – y – 3 z = 6 D C Pick another pair of equations to eliminate the x–term. A and B 2 x – y + 3 z = – 4 – 2 x – 4 y + 10 z = – 22 2 x – y + 3 z = – 4 A – 2(x x ++ 2 y 2 y –– 5 z 5 z)==11 – 2(11) B – 5 y + 13 z = – 26 E +1 Label the new equation E. – 5(––y y–– 3 z) 3 z == – 5(6) 6 – 5 y + 13 z = – 26 x + 3 y – 2 z = 5 +1 x + 3(0) – 2(– 2) = 5 x+0+4=5 x = 1 +1 D E +1 5 y + 15 z = – 30 +1 – y – 3(– 2) = 6 – 5 y + 13 z = – 26 +1 – y + 6 = 6 –y=0 28 z = – 56 y=0 z = – 2 +1 (1, 0, – 2) +1 +1
Yesterday we solved systems of equations in 3 variables. Today we are going to look at the cases where there is no solution or infinitely many solutions. Infinitely Many Solutions No Solution planes intersect in a line planes intersect in the same planes have no point in common
Example #1: Infinitely Many Solutions Solve the system of equations. 4 x – 6 y + 4 z = 12 6 x – 9 y + 6 z = 18 A B 5 x – 8 y + 10 z = 20 C Eliminate x using A and B. 3(4 x 4 x –– 6 y 6 y ++ 4 z 4 z)==12 3(12) A 12 x – 18 y + 12 z = 36 – 2(6 x 6 x –– 9 y 9 y ++ 6 z 6 z)==18 – 2(18) B – 12 x + 18 y – 12 z = – 36 0=0 The equation 0 = 0 is always true. This indicates that the first two equations represent the same plane. ALWAYS check to see if this plane intersects the third plane.
Example #1: Infinitely Many Solutions Solve the system of equations. 4 x – 6 y + 4 z = 12 6 x – 9 y + 6 z = 18 A B 5 x – 8 y + 10 z = 20 C Eliminate x using A and B. 3(4 x 4 x –– 6 y 6 y ++ 4 z 4 z)==12 3(12) A 12 x – 18 y + 12 z = 36 – 2(6 x 6 x –– 9 y 9 y ++ 6 z 6 z)==18 – 2(18) B – 12 x + 18 y – 12 z = – 36 0=0 4 x 5(4 x – 6 y + 4 z) = 12 = 5(12) A 20 x – 30 y + 20 z = 60 – 20 x + 32 y – 40 z = – 80 – 4(5 x 5 x – 8 y + 10 z) = 20 = – 4(20) C 2 y – 20 z = – 20 Since we have an equation, the planes interest in the line. Therefore, there are infinitely many number of solutions.
Example #2: No Solution Solve the system of equations. 6 x + 12 y – 8 z = 24 A 9 x + 18 y – 12 z = 30 B 4 x + 8 y – 7 z = 26 Eliminate x using A and B. 6 x 3(6 x + 12 y – 8 z) = 24 = 3(24)A – 2(9 x 9 x ++18 y 18 y–– 12 z)==30 – 2(30) B C 18 x + 36 y – 24 z = 72 – 18 x – 36 y + 24 z = – 60 0 = 12 The equation 0 = 12 is never true. Therefore, there is no solution of this system.
Solve each equation for the variable. Be sure to check for extraneous solutions. Recall: An extraneous solution is a solution of the simplified form of an equation that does not satisfy the original equation. For example, sometimes we had to eliminate answers when we solved logarithmic equations. Example #1: Step #1: Leave the radicals isolated on both sides. Step #2: Eliminate the radicals by squaring both sides. Step #3: Solve for x.
Example #2: Step #1: Leave the radical isolated on the right hand side. Step #2: Eliminate the radical by squaring both sides. Step #3: Solve for x. Check: CHECK!!! √ Does NOT work.
Example #3: Step #1: Leave the radical isolated on left hand side. Step #2: Eliminate the radical by squaring both sides. Step #3: Solve for x. Check: CHECK!!! Check: √ Does NOT work.
Complete the worksheets and the CST practice.
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