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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.

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

${:(x+2y+3z = ,-5),(3x+y-3z = , 4),(-3x+4y+7z = , -7):}$
1. $(-1,0,-2)$
2. $(1,1,2)$
3. $(-1,1,-2)$
4. $(0,1,2)$
3.
Solve.
$x + 2y + z = -4$
$x - y - z = -1$
$-5x + 3y + 4z = 4$
1. $(-1,-1,-1)$
2. $(5,3,3)$
3. $(3,-5,1)$
4. $(-5, 5, -9)$
4.
Solve.
$2x + 2y + 2z = 18$
$3x + 5y + 4z = 22$
$x + 4y + 2z = 12$
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.
$3x+2y-z=13$
$2x-y+2z=6$
$x+y+z=3$
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. $x + y + z = 22, \ \ \ 2x + 2y + 2z = 20, \ \ \ 2x + y + z = 20; \ \ \ (x,y,z) = (6, 5,3)$
2. $2x+y+2z=22, \ \ \ x+2y+2z=20, \ \ \ x+2y+z+20; \ \ \ (x,y,z) = (5,6,3)$
3. $2x+2y+z=22, \ \ \ x+y+2x=20, \ \ \ x+3y+z=20; \ \ \ (x,y,z) = (5,3,6)$
4. $x+y+2z=22, \ \ \ 2x+z+2z=20, \ \ \ x+2y+2z=20; \ \ \ (x,y,z) = (3,6,5)$
7.
Solve the system.
$x-2y+z=8$
$y-z=4$
$z=3$

8.
Solve the system.
$x+y+z=6$
$x=2y$
$z=x+1$

9.
Solve the system.
$3x+y-z=15$
$x-y+3z=-19$
$2x+2y+z=4$

10.
Solve the system.
$x+y+3z= 20$
$2x+4y+5z= 30$
$x+y+4z=26$

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