how can i solve these equations?
x + 3y – z = 2
x – 2y + 3z = 7
x + 2y – 5z = –21
We eliminate one of the unknowns (x, y or z, take your choice),
leaving two unknowns. Then, we eliminate a second one, giving us
an equation with only one of the unknowns. Solve for that and
plug that value into one of the equations to solve for a second
unknown. Finally plug both of those values into an equation to
solve for the third unknown. It sounds complicated, but if you
follow a logical sequence, the problem solves itself.
1) x + 3y – z = 2
2) x – 2y + 3z = 7
3) x + 2y – 5z = –21
If we subtract equation 3 from equation 2, we eliminate the x.
x – 2y + 3z = 7
-(x + 2y – 5z = –21)
------------------------
- 4y + 8z = 28
4) -4y + 8z = 28
Subtract equation 1 from equation 2, eliminating the x again.
x – 2y + 3z = 7
-(x + 3y – z = 2)
----------------------
- 5y + 4z = 5
5) -5y + 4z = 5
You now have two equations with only a y and a z. The easiest
step now is to eliminate the z. Multiply equation 5 by 2.
2 * (-5y + 4z) = 5 * 2
6) -10y + 8z = 10
Subtract equation 6 from equation 4, eliminating the z.
-4y + 8z = 28
-(-10y + 8z = 10)
---------------------
6y = 18
6y = 18
y = 3 <<<<<<<<<<<<<<<<<<<<<
Plug that into equation 5 to solve for z.
-5y + 4z = 5
-5(3) + 4z = 5
-15 + 4z = 5
4z = 20
z = 5 <<<<<<<<<<<<<<<<<<<<<
Plug the values of y and z into equation 1 to solve for x.
x + 3y – z = 2
x + 3(3) – 5 = 2
x + 9 - 5 = 2
x + 4 = 2
x = -2 <<<<<<<<<<<<<<<<<<<<<
Always check the answers by plugging all three values
into one of the original equations. Using all three would
be even better.
Equation 2:
x – 2y + 3z = 7
(-2) – 2(3) + 3(5) = 7
-2 - 6 + 15 = 7
-8 + 15 = 7
7 = 7
Equation 3:
x + 2y – 5z = –21
(-2) + 2(3) – 5(5) = –21
-2 + 6 - 25 = -21
6 - 27 = -21
-21 = -21
Answer: x = -2, y = 3, z = 5