3x + 6y – 6z = 9 2x – 5y + 4z = 6 -x +16y + 14z = -3 what is the answer
how do you solve it and what are the answers ?
1) 3x + 6y – 6z = 9
2) 2x – 5y + 4z = 6
3) -x +16y + 14z = -3
The first objective is to eliminate z so we can
solve for x and y. It's a multi-step process, so
follow along.
Multiply equation one by 4.
4 * (3x + 6y – 6z) = 9 * 4
4) 12x + 24y - 24z = 36
Multiply equation 2 by 6.
6 * (2x – 5y + 4z) = 6 * 6
5) 12x - 30y + 24z = 36
Now, we have two equations with a "24z" term. Add the
equations and the z drops out.
Add equation five to equation four.
12x + 24y - 24z = 36
+(12x - 30y + 24z = 36)
----------------------------------
24x - 6y = 72
6) 24x - 6y = 72
The same process applies to equations two and three.
Multiply equation two by 7 this time.
7 * (2x – 5y + 4z) = 6 * 7
7) 14x - 35y + 28z = 42
Multiply equation three by 2.
2 * (-x + 16y + 14z) = -3 * 2
8) -2x + 32y + 28z = -6
Subtract equation eight from equation seven.
14x - 35y + 28z = 42
-(-2x + 32y + 28z = -6)
---------------------------------
16x - 67y = 48
9) 16x - 67y = 48
Looking at equations six and nine, it would be simpler
to eliminate the x. The multipliers are smaller.
Multiply equation six by 2.
2 * (24x - 6y) = 72 * 2
10) 48x - 12y = 144
Multiply equation nine by 3.
3 * (16x - 67y) = 48 * 3
11) 48x - 201y = 144
Subtract equation eleven from equation 10.
48x - 12y = 144
-(48x - 201y = 144)
---------------------------
189y = 0
189y = 0
y = 0 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
Substitute that value into equations six and nine
to solve for x and verify.
Six:
24x - 6y = 72
24x - 6(0) = 72
24x - 0 = 72
24x = 72
x = 3 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
Nine:
16x - 67y = 48
16x - 67(0) = 48
16x - 0 = 48
16x = 48
x = 3 same answer for x
To solve for z,substitute the x and y values into
the three original equations.
One:
3x + 6y – 6z = 9
3(3) + 6(0) – 6z = 9
9 + 0 - 6z = 9
9 - 6z = 9
-6z = 9 - 9
-6z = 0
z = 0 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
Two:
2x – 5y + 4z = 6
2(3) – 5(0) + 4z = 6
6 - 0 + 4z = 6
6 + 4z = 6
4z = 6 - 6
4z = 0
z = 0 same answer
Three:
-x +16y + 14z = -3
-(3) +16(0) + 14z = -3
-3 + 0 + 14z = -3
-3 + 14z = -3
14z = -3 + 3
14z = 0
z = 0 once again, same answer
x = 3, y = 0, z = 0