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Systems of Linear Equations in Three Variables (Grades 11-12)

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Systems of Linear Equations in Three Variables

1. 
Solve the system of equations.

[math]2x-y+2z=15[/math]
[math]-x+y+z=3[/math]
[math]3x-y+2z=18[/math]
  1. [math](5,3,1)[/math]
  2. [math](3,1,5)[/math]
  3. [math](-3,5,1)[/math]
  4. [math]"No Solution"[/math]
2. 
What is the solution to the following system?

[math]{:(x+2y+3z = ,-5),(3x+y-3z = , 4),(-3x+4y+7z = , -7):}[/math]
  1. [math](-1,0,-2)[/math]
  2. [math](1,1,2)[/math]
  3. [math](-1,1,-2)[/math]
  4. [math](0,1,2)[/math]
3. 
Solve.
[math] x + 2y + z = -4 [/math]
[math] x - y - z = -1 [/math]
[math] -5x + 3y + 4z = 4 [/math]
  1. [math](-1,-1,-1)[/math]
  2. [math](5,3,3)[/math]
  3. [math](3,-5,1)[/math]
  4. [math](-5, 5, -9)[/math]
4. 
Solve.
[math]2x + 2y + 2z = 18[/math]
[math]3x + 5y + 4z = 22[/math]
[math]x + 4y + 2z = 12[/math]
  1. x = 38, y = 16, z = -37
  2. x = 12, y = 8, z = -11
  3. x = 22, y = 8, z = -21
  4. x = -20, y = -8, z =-21
5. 
[math]3x+2y-z=13[/math]
[math]2x-y+2z=6[/math]
[math]x+y+z=3[/math]
  1. x = 0 y = 4 z = -1
  2. x = 4 y = 0 z = -1
  3. x = -1 y = 4 z = 0
  4. x = 1 y = -4 z = 0
6. 
Kevin weighs sets of small rock samples for his science class. A set of 2 quartz, 2 mica, and 1 granite rocks weigh 22 grams. A set of 1 quartz, 1 mica, and 2 granite rocks weigh 20 grams. A set of 1 quartz, 3 mica, and 1 granite rocks weigh 20 grams. Samples of each rock type have the same weight. Write a system of linear equations. Solve the system to determine the weight of each rock. x = weight on the quartz rock, y = the weight of the mica rock, and z = the weight of the granite rock.
  1. [math]x + y + z = 22, \ \ \ 2x + 2y + 2z = 20, \ \ \ 2x + y + z = 20; \ \ \ (x,y,z) = (6, 5,3)[/math]
  2. [math]2x+y+2z=22, \ \ \ x+2y+2z=20, \ \ \ x+2y+z+20; \ \ \ (x,y,z) = (5,6,3)[/math]
  3. [math]2x+2y+z=22, \ \ \ x+y+2x=20, \ \ \ x+3y+z=20; \ \ \ (x,y,z) = (5,3,6)[/math]
  4. [math]x+y+2z=22, \ \ \ 2x+z+2z=20, \ \ \ x+2y+2z=20; \ \ \ (x,y,z) = (3,6,5)[/math]
7. 
Solve the system.
[math]x-2y+z=8[/math]
[math]y-z=4[/math]
[math]z=3[/math]



8. 
Solve the system.
[math]x+y+z=6[/math]
[math]x=2y[/math]
[math]z=x+1[/math]





9. 
Solve the system.
[math]3x+y-z=15[/math]
[math]x-y+3z=-19[/math]
[math]2x+2y+z=4[/math]





10. 
Solve the system.
[math]x+y+3z= 20[/math]
[math]2x+4y+5z= 30[/math]
[math]x+y+4z=26[/math]










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