Math  /  Algebra

QuestionSolve the system. Give your answer as (x,y,z)(x, y, z) 3x4y4z=306x4y+5z=246x+y+3z=2\begin{array}{l} 3 x-4 y-4 z=-30 \\ -6 x-4 y+5 z=24 \\ 6 x+y+3 z=2 \end{array} (x,y,z)=(x, y, z)=

Studdy Solution

STEP 1

1. We are solving a system of three linear equations with three variables: xx, yy, and zz.
2. The solution will be a single ordered triple (x,y,z)(x, y, z).
3. We will use the elimination method to solve this system.

STEP 2

1. Eliminate one variable from two pairs of equations.
2. Solve the resulting system of two equations with two variables.
3. Substitute back to find the third variable.
4. Verify the solution by substituting back into the original equations.

STEP 3

Choose two pairs of equations and eliminate the same variable from each pair. Let's eliminate xx.
First, consider equations 1 and 2: \[ \begin{array}{l} 3x - 4y - 4z = -30 \\ -6x - 4y + 5z = 24 \end{array}$
Multiply the first equation by 2: \[ 6x - 8y - 8z = -60 $
Now add it to the second equation: \[ (6x - 8y - 8z) + (-6x - 4y + 5z) = -60 + 24 $
Simplify: \[ -12y - 3z = -36 $
Divide by -3: \[ 4y + z = 12 $

STEP 4

Now consider equations 2 and 3: \[ \begin{array}{l} -6x - 4y + 5z = 24 \\ 6x + y + 3z = 2 \end{array}$
Add these equations together: \[ (-6x - 4y + 5z) + (6x + y + 3z) = 24 + 2 $
Simplify: \[ -3y + 8z = 26 $

STEP 5

Now solve the system of two equations: \[ \begin{array}{l} 4y + z = 12 \\ -3y + 8z = 26 \end{array}$
From the first equation, express zz in terms of yy: \[ z = 12 - 4y $
Substitute into the second equation: \[ -3y + 8(12 - 4y) = 26 $
Simplify: \[ -3y + 96 - 32y = 26 $
Combine like terms: \[ -35y + 96 = 26 $
Subtract 96 from both sides: \[ -35y = -70 $
Divide by -35: \[ y = 2 $

STEP 6

Substitute y=2y = 2 back into the equation z=124yz = 12 - 4y: \[ z = 12 - 4(2) $
Simplify: \[ z = 4 $

STEP 7

Substitute y=2y = 2 and z=4z = 4 into one of the original equations to solve for xx. Use the third equation: \[ 6x + y + 3z = 2 $
Substitute: \[ 6x + 2 + 3(4) = 2 $
Simplify: \[ 6x + 2 + 12 = 2 $
Combine like terms: \[ 6x + 14 = 2 $
Subtract 14 from both sides: \[ 6x = -12 $
Divide by 6: \[ x = -2 $

STEP 8

Verify the solution (2,2,4)(-2, 2, 4) by substituting back into the original equations.
Equation 1: \[ 3(-2) - 4(2) - 4(4) = -30 $
Simplify: \[ -6 - 8 - 16 = -30 $
True.
Equation 2: \[ -6(-2) - 4(2) + 5(4) = 24 $
Simplify: \[ 12 - 8 + 20 = 24 $
True.
Equation 3: \[ 6(-2) + 2 + 3(4) = 2 $
Simplify: \[ -12 + 2 + 12 = 2 $
True.
The solution is verified.
The solution is (2,2,4)(-2, 2, 4).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord