solve the system: 5x-10y+4z=-73; -x+2y-3z=19; 4x-3y=5z=-42
1) 5x - 10y + 4z = -73
2) -x + 2y - 3z = 19
3) 4x - 3y + 5z = -42
Equation three had "=5z" so I changed the equal sign
to a plus sign. It happens a lot; the shift key doesn't
register, so what was supposed to be "+" comes out "=".
We're going to eliminate z so we can work on finding x and y.
We need the coefficients of z to be equal (disregarding the sign).
Equation one; multiply by 3:
3 * (5x - 10y + 4z) = -73 * 3
4) 15x - 30y + 12z = -219
Equation two; multiply by 4:
4 * (-x + 2y - 3z) = 19 * 4
5) -4x + 8y - 12z = 76
Add equation five to equation four, eliminating z.
15x - 30y + 12z = -219
+(-4x + 8y - 12z = 76)
-----------------------------------
11x - 22y = -143
6) 11x - 22y = -143
Follow the same procedure with equations two and three.
Equation two: multiply by 5 this time:
5 * (-x + 2y - 3z) = 19 * 5
7) -5x + 10y - 15z = 95
Equation three: multiply by 3:
3 * (4x - 3y + 5z) = -42 * 3
8) 12x - 9y + 15z = -126
Add equation eight to equation seven, eliminating z.
-5x + 10y - 15z = 95
+(12x - 9y + 15z = -126)
------------------------------------
7x + y = -31
9) 7x + y = -31
Multiply equation nine by 22; then we will add equation six.
22 * (7x + y) = -31 * 22
154x + 22y = -682
+(11x - 22y = -143)
---------------------------
165x = -825
x = -5 <<<<<<<<<<<<<<<<<<<<<<<<<<
Substitue that into equations six and nine to find y. They
will serve to verify the calculations if you do it with two equations.
11x - 22y = -143
11(-5) - 22y = -143
-55 - 22y = -143
-22y = -143 + 55
-22y = -88
y = 4 <<<<<<<<<<<<<<<<<<<<<<<<<<
7x + y = -31
7(-5) + y = -31
-35 + y = -31
y = -31 + 35
y = 4 same value for y
Now, we can go back to the original equations, plug in
both x and y, and solve for z. To check and verify, we
will use all three of the original equations.
One:
5x - 10y + 4z = -73
5(-5) - 10(4) + 4z = -73
-25 - 40 + 4z = -73
-65 + 4z = -73
4z = -73 + 65
4z = -8
z = -2 <<<<<<<<<<<<<<<<<<<<<<<<<<
Two:
-x + 2y - 3z = 19
-(-5) + 2(4) - 3z = 19
5 + 8 - 3z = 19
13 - 3z = 19
-3z = 19 - 13
-3z = 6
z = -2 same answer for z
Three:
4x - 3y + 5z = -42
4(-5) - 3(4) + 5z = -42
-20 - 12 + 5z = -42
-32 + 5z = -42
5z = -42 + 32
5z = -10
z = -2 again, verified
x = -5, y = 4, z = -2