Problem: x - z = -3, y + z = 9, -2x + 3y +5z = 33
1) x - z = -3
2) y + z = 9
3) -2x + 3y + 5z = 33
Add equation 2 to equation 1.
x - z = -3
+( y + z = 9)
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x + y = 6
4) x + y = 6
Multiply equation 2 by 5.
5(y + z) = 9 * 5
5) 5y + 5z = 45
Subtract equation 3 from equation 5.
5y + 5z = 45
-(-2x + 3y + 5z = 33)
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2x + 2y = 12
6) 2x + 2y = 12
Multiply equation 4 by 2.
2(x + y) = 6 * 2
7) 2x + 2y = 12
Subtract equation 7 from equation 6.
2x + 2y = 12
-(2x + 2y = 12)
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0x = 0 We are solving for x, not y
x = 0 <<<<<<<<<<<<<<<<<<<
Use equation 4 to solve for y.
x + y = 6
0 + y = 6
y = 6 <<<<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
-2x + 3y + 5z = 33
-2(0) + 3(6) + 5z = 33
0 + 18 + 5z = 33
5z = 15
z = 3 <<<<<<<<<<<<<<<<<<<
Check the values.
1) x - z = -3
0 - 3 = -3
-3 = -3
2) y + z = 9
6 + 3 = 9
9 = 9
3) -2x + 3y + 5z = 33
-2(0) + 3(6) + 5(3) = 33
0 + 18 + 15 = 33
33 = 33
Those numbers work.....
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What if we had solved for y instead of x here:
2x + 2y = 12
-(2x + 2y = 12)
-------------------
0y = 0 We are solving for y, not x
y = 0 <<<<<<<<<<<<<<<<<<<
Use equation 4 to solve for x.
x + y = 6
x + 0 = 6
x = 6 <<<<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
-2x + 3y + 5z = 33
-2(6) + 3(0) + 5z = 33
-12 + 0 + 5z = 33
5z = 45
z = 9 <<<<<<<<<<<<<<<<<<<
Check the values.
1) x - z = -3
6 - 9 = -3
-3 = -3
2) y + z = 9
0 + 9 = 9
9 = 9
3) -2x + 3y + 5z = 33
-2(6) + 3(0) + 5(9) = 33
-12 + 0 + 45 = 33
33 = 33
Those numbers work, too.
That complicates the solution.
Answer 1: x = 0, y = 6, z = 3
Answer 2: x = 6, y = 0, z = 9