what is the answer to 3x+y+z=14, -x+2y+-3z=-9, 5x+-y+z=30
solve the system using the elimination method 3x+y+z=14 -x+2y+-3z=-9 5x+-y+z=30
1) 3x + y + z = 14
2) -x + 2y - 3z = -9
3) 5x - y + z = 30
We don't need to say "plus a negative number." Just subtract the number.
Add equation 3 to equation 1.
3x + y + z = 14
+(5x - y + z = 30)
---------------------
8x + 2z = 44
4) 8x + 2z = 44
Multiply equation 3 by 2.
2 * (5x - y + z) = 30 * 2
5) 10x - 2y + 2z = 60
Add equation 2 to equation 5.
10x - 2y + 2z = 60
+(-x + 2y - 3z = -9)
-----------------------
9x - z = 51
6) 9x - z = 51
Multiply equation 6 by 2.
2 *(9x - z) = 51 * 2
7) 18x - 2z = 102
Add equation 7 to equation 4.
8x + 2z = 44
+(18x - 2z = 102)
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26x = 146
26x = 146
x = 146/26
x = 73/13 <<<<<<<<<<<<<<<<<<<<<<
Plug that into equation 6 to solve for z.
9x - z = 51
9(73/13) - z = 51
657/13 - z = 51
-z = 51 - 657/13
-z = 663/13 - 657/13
-z = 6/13
z = -6/13 <<<<<<<<<<<<<<<<<<<<<<
Plug those into equation 1 to solve for y.
3x + y + z = 14
3(73/13) + y - 6/13 = 14
219/13 + y - 6/13 = 14
213/13 + y = 14
y = 14 - 213/13
y = 182/13 - 213/13
y = -31/13 <<<<<<<<<<<<<<<<<<<<<<
Use equation 3 to check the answers.
5x - y + z = 30
5(73/13) - (-31/13) - 6/13 = 30
365/13 + 31/13 - 6/13 = 30
390/13 = 30
30 = 30
While we're at it, use equation 2 to check the answers.
-x + 2y - 3z = -9
-(73/13) + 2(-31/13) - 3(-6/13) = -9
-73/13 - 62/13 + 18/13 = -9
-117/13 = -9
-9 = -9
We have x = 73/13, y = -31/13 and z = -6/13