Question

Solve the system by substitution.
-x-y-z = -8
- 4x + 4y + 5z = 7
2x + 2z = 4

Please show all steps.

Answer 1

Answer:

The solution set of given equations -x-y-z = -8 and - 4x + 4y + 5z = 7 and 2x + 2z = 4  is (3, 6, -1)

Solution:

Given, set linear equations are

-x – y – z = -8 ⇒ x + y + z = 8 → (1)

-4x + 4y + 5z = 7 ⇒ 4x – 4y – 5z = -7 → (2)

2x + 2z = 4 ⇒ x + z = 2 → (3)

We have to solve the above given equations using substitution method.

Now take (3), x + z = 2 ⇒ x = 2 – z  

So substitute x value in (1)  

(1) ⇒ (2 – z) + y + z = 8 ⇒ 2 + y + z – z = 8 ⇒ y + 0 = 8 – 2 ⇒ y = 6.

Now substitute x and y values in (2)

(2) ⇒ 4(2 – z) – 4(6) – 5z = - 7 ⇒ 8 – 4z – 24 – 5z = -7 ⇒ -9z – 16 = -7 ⇒ 9z = 7 – 16 ⇒ 9z = -9 ⇒ z = -1

Now substitute z value in (3)

(3) ⇒ x – 1 = 2 ⇒ x = 2 + 1 ⇒ x = 3

Hence, the solution set of given equations is (3, 6, -1).

Answer 2

Using the substitution method, the solution to the system of equations given is: (3, 6, -1).

Given the system:

-x-y-z = -8 --> eqn. 1

- 4x + 4y + 5z = 7 --> eqn. 2

2x + 2z = 4 --> eqn. 3

Rewrite eqn. 3 to make x the subject of the formula.

Thus:

$2x + 2z = 4 \\\\2x = 4 - 2z$

• Divide both sides by 2

$x = 2 - z$

Substitute x = 2 - z into eqn. 1 to find y.

$-(2 - z)-y-z = -8 \\\\-2 + z - y - z = - 8$

• Add like terms

$-2 + z - y - z = - 8\\\\-2 - y = - 8\\\\-y = -8 + 2\\\\-y = -6\\\\\mathbf{y = 6}$

Substitute x = 2 - z and y = 6 into eqn. 2 to find z

$- 4(2 - z) + 4(6) + 5z = 7\\\\-8 + 4z + 24 + 5z = 7$

• Add like terms

$-8 + 4z + 24 + 5z = 7\\\\9z + 16 = 7\\\\9z = 7 - 16\\\\9z = -9\\\\\mathbf{z = -1}$

Substitute z = -1 into eqn. 3 to find x.

$2x + 2(-1) = 4\\\\2x - 2 = 4\\\\2x = 4 + 2\\\\2x = 6\\\\\mathbf{x = 3}$

Therefore, using the substitution method, the solution to the system of equations given is: (3, 6, -1).

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