solve a system of linear equations with three variables:
x+y-3z=1
2x-y+z=9
3x+y-4z=8
1) x + y - 3z = 1
2) 2x - y + z = 9
3) 3x + y - 4z = 8
Multiply equation 2 by 3.
3(2x - y + z) = 9 * 3
4) 6x - 3y + 3z = 27
Add equation 4 to equation 1.
x + y - 3z = 1
+(6x - 3y + 3z = 27)
--------------------------
7x - 2y = 28
5) 7x - 2y = 28
Multiply equation 2 by 4.
4(2x - y + z) = 9 * 4
6) 8x - 4y + 4z = 36
Add equation 3 to equation 6.
8x - 4y + 4z = 36
+(3x + y - 4z = 8)
------------------------
11x - 3y = 44
7) 11x - 3y = 44
Multiply equation 5 by 3.
3(7x - 2y) = 28 * 3
8) 21x - 6y = 84
Multiply equation 7 by 2.
2(11x - 3y) = 44 * 2
9) 22x - 6y = 88
Subtract equation 9 from equation 8.
21x - 6y = 84
-(22x - 6y = 88)
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-x = -4
x = 4 <<<<<<<<<<<<<<<<<
Use equation 5 to solve for y.
7x - 2y = 28
7(4) - 2y = 28
28 - 2y = 28
-2y = 0
y = 0 <<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
3x + y - 4z = 8
3(4) + 0 - 4z = 8
12 - 4z = 8
-4z = -4
z = 1 <<<<<<<<<<<<<<<<<
Check your answers.
1) x + y - 3z = 1
4 + 0 - 3(1) = 1
4 - 3 = 1
1 = 1
2) 2x - y + z = 9
2(4) - 0 + 1 = 9
8 + 1 = 9
9 = 9
3) 3x + y - 4z = 8
3(4) + 0 - 4(1) = 8
12 - 4 = 8
8 = 8
Everything checks.
Answer: x = 4, y = 0, z = 1