Solve the system using the elimination method
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2 Answers

2x + 4y + 3z = -11

4x - 3y + 4z = 34

x + 2y - 4z = -33

multiply the top equation by 4, the middle by -3, and the bottom by 3

8x + 16y + 12z = -44

-12x + 9y - 12z = -102

3x + 6y - 12z = -99

add the top equation to the middle and bottom

8x + 16y + 12z = -44

-4x + 25y + 0z = -146

11x + 22y + 0z = -143

multiply the middle equation by 22 and the bottom by -25

8x + 16y + 12z = -44

-88x + 550y + 0z = -3212

-275x - 550y + 0z = 3575

add the middle equation to the bottom

8x + 16y + 12z = -44

-88x + 550y + 0z = -3212

-363x + 0y + 0z = 363

divide the bottom equation by -363

8x + 16y + 12z = -44

-88x + 550y + 0z = -3212

x + 0y + 0z = -1

Now we know that x = -1

multiply the bottom equation by 88

8x + 16y + 12z = -44

-88x + 550y + 0z = -3212

88x + 0y + 0z = -88

add the bottom equation to the middle

8x + 16y + 12z = -44

0x + 550y + 0z = -3300

88x + 0y + 0z = -88

divide the middle equation by 550

8x + 16y + 12z = -44

0x + y + 0z = -6

88x + 0y + 0z = -88

Now we know that y = -6

multiply the middle equation by -16 and divide the bottom equation by -11

8x + 16y + 12z = -44

0x - 16y + 0z = 96

-8x + 0y + 0z = 8

add the middle and bottom equations to the top

0x + 0y + 12z = 60

0x - 16y + 0z = 96

-8x + 0y + 0z = 8

divide the top equation by 12

0x + 0y + z = 5

0x - 16y + 0z = 96

-8x + 0y + 0z = 8

Now we know that z = 5

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Answer:  x = -1, y = -6, z = 5
by Level 13 User (103k points)
If we double the third equation we get 2x + 4y - 8z = -66. So we can eliminate x and y between the first and this new equation, simply subtract one from the other. Easier to subtract from the first equation to give us 11z = 55, so z = 5. Now substitute this value into equations 1 and 2. 2x + 4y = -26 and 4x - 3y = 14. Double the first of these and we get 4x + 8y = -52. Subtract the other equation from it and we get 11y = -66, so y = -6. Take any equation and substitute for y and z. Let's take the last equation x - 12 - 20 = -33, from which x = -1. Try the values out in each original equation to make sure they're right. In problems of this sort try to make the work easy by spotting coincidences, like the fact that the first and third equations eliminate two variables at once because of a simple multiple, 2 in this case.
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