Linear systems of equations with 4 unknowns <mtable columnalign="right left right left right le

tinydancer27br

tinydancer27br

Answered question

2022-05-26

Linear systems of equations with 4 unknowns
7 x + 4 y + 3 z + 2 w = 46 5 x y + 4 w = 23 x + z = 6 3 x + 7 w = 15

Answer & Explanation

Smarmabagd7

Smarmabagd7

Beginner2022-05-27Added 6 answers

With z = 6 x we can eliminate z so we get:
7 x + 4 y + 3 ( 6 x ) + 2 w = 46
Then distributing the 3 3:
7 x + 4 y + 18 3 x + 2 w = 46
Moving variable terms to left of the left side and constant term to right of left side then combining like terms we get:
4 x + 4 y + 2 w + 18 = 46
Subtracting 18 from both sides so we get
4 x + 4 y + 2 w + 18 = 46
Dividing by 2, then it becomes:
5 x y + 4 w = 23
5 x y + 4 w = 23
3 x + 7 w = 15
Then deriving from the first equation we get
w = 14 2 x 2 y
Substituting w we get:
5 x y + 4 ( 14 2 x 2 y ) = 23
3 x + 7 ( 14 2 x 2 y ) = 15
By the process the steps for the first one are below:
5 x y + 56 8 x 8 y = 23
5 x y 8 x 8 y + 56 = 23
3 x 9 y + 56 = 23
3 x 9 y = 33
Let's get rid of the negatives:
3 x + 9 y = 33
And the steps for the second one are below:
3 x + 98 14 x 14 y = 15
11 x 14 y + 98 = 15
11 x 14 y = 83
Get rid of the negatives, there are the 2 equations:
3 x + 9 y = 33
11 x + 14 y = 83
We could do elimination here:
11 ( 3 x + 9 y ) = 11 × 33
3 ( 11 x + 14 y ) = 3 × 83
We get then:
33 x + 99 y = 363
33 x + 42 y = 249
Subtracting one from the other we get:
( 33 x + 99 y ) ( 33 x + 42 y ) = 363 249
We can distribute the minus sign on the left side:
33 x + 99 y 33 x 42 y = 363 249
Simplifying we get:
57 y = 114
Dividing both sides by 57 we get:
y = 2
Now let's plug this in into any one of the equations before multiplying (in this case, I used 3 x + 9 y = 33 just for simplicity, simplifying, through the steps:
3 x + 9 × 2 = 33
3 x + 18 = 33
3 x = 15
x = 5
We've got x = 5 and y = 2. Can you finish the rest to get z = 1 and w = 0?

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