Solve this system of equations?

Question

x+9y-6z= -2
-2x-8y-7x=87
-10x+4y-5z=5

Answer 1

You can do this using matrices, but I'm not sure how well you know about those, so we'll do this by keeping all of the variables in place.

I'm not sure if you mistyped the second equation - you have "7x" where it might be "7z." I'll solve it as if it were "7z" first.

Take the first equation, multiply by 2, and add to the second one:

2x + 18y - 12z = -4
-2x - 8y - 7z = 87
-------------------------
10y - 19z = 83

Solve for y:

10y - 19z = 83
10y = 19z + 83
y = 1.9z + 8.3

Multiply the first quation by 10 and add it to the third:
10x + 90y - 60z = -20
-10x + 4y - 5z = 5
-------------------------
94y - 65z = -15

Plug in the first combined equation (y = 1.9z + 8.3) and solve for z:
94(1.9z + 8.3) - 65z = -15
178.6z + 780.2 - 65z = -15
113.6z + 780.2 = - 15
113.6z = -795.2
z = -7

Solve for y:
y = 1.9z + 8.3
y = 1.9(-7) + 8.3
y = -5

Finally, pick an equation and solve for x:
x + 9y - 6z = -2
x + 9(-5) - 6(-7) = -2
x - 45 + 42 = -2
x - 3 = -2
x = 1

Thus, x = 1, y = -5, and z = -7.

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If your second equation is really -2x - 8y - 7x = 87, we can combine the x terms and come up with -9x - 8y = 87. Then we can do some similar stuff as above to solve the system.

In the first case, we eliminated the x terms, but since our second equation doesn't have a z term, we can work to eliminate that one instead. Multiply the first equation by -5 and the second by 6 and add them together:

-5x - 45y + 30z = 10
-60x + 24y - 30z = 30
-------------------------
-65x - 21y = 40

We can take our seond equation and solve for x:

-9x - 8y = 87
-9x = 8y + 87
x = -(8y + 87)/9

We can then take this and plug it in to our combined equation:

-65x - 21y = 40
-65(-(8y + 87)/9) - 21y = 40

You can see that this can get pretty complicated with the fractions and all, so my bet is that your second equation should read "7z." But if not, simply solve this one here for y, plug in and solve for x, and then plug both in to solve for z.