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Solve the system.
3x + 2y + 4z = 11
2x – y + 3z = 4
5x – 3y + 5z = -1
Enter your answer as an ordered triple.


Someone help please!!

Solve The System 3x 2y 4z 11 2x Y 3z 4 5x 3y 5z 1 Enter Your Answer As An Ordered Triple Someone Help Please class=

Sagot :

My math problem is write one hundredths as a decimal

Answer:   (x, y, z) = (-3, 2, 4)

In other words, x = -3, y = 2 and z = 4.

You'll enter the number only into each box.

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Explanation:

There are a few routes we can take to solve this. I'll use substitution.

First let's isolate y in the second equation.

2x - y + 3z = 4

2x + 3z = 4+y

4+y = 2x+3z

y = 2x+3z-4

This is then plugged into the first equation

3x+2y+4z = 11

3x+2( y )+4z = 11

3x+2( 2x+3z-4)+4z = 11 .... replace y with 2x+3z-4; since y = 2x+3z-4

3x+4x+6z-8+4z =11

7x+10z-8 = 11

7x+10z = 11+8

7x+10z = 19 .... we'll call this equation (4) and come back to this later

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Move onto the third equation (of the original equations given)

Plug in y = 2x+3z-4

5x - 3y + 5z = -1

5x - 3( y ) + 5z = -1

5x - 3( 2x+3z-4 ) + 5z = -1

5x - 6x - 9z + 12 + 5z = -1

-x - 4z + 12 = -1

-x - 4z = -1-12

-x - 4z = -13 ..... let's call this equation (5)

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To recap, we found equation (4) and equation (5) to be

  • 7x+10z = 19
  • -x - 4z = -13

We can then solve for x in equation (5) like so

-x - 4z = -13

-x = -13 + 4z

-x = 4z - 13

x = -(4z - 13)

x = -4z + 13

This is then plugged into equation (4)

7x+10z = 19

7(-4z+13)+10z = 19 .... note how we now have one variable

-28z+91+10z = 19

-18z+91 = 19

-18z = 19-91

-18z = -72

z = -72/(-18)

z = 4

We can use this value of z to find x

x = -4z+13

x = -4(4)+13

x = -16+13

x = -3

Use the values of x and z to find y. We'll revisit the first equation in which we isolated y

y = 2x+3z-4

y = 2(-3)+3(4)-4

y = -6+12-4

y = 6-4

y = 2

So overall, the solution to this system is (x, y, z) = (-3, 2, 4)

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Checking the answer:

We'll replace x,y,z with the values we found.

  • In the first equation, 3x+2y+4z = 11 becomes 3(-3)+2(2)+4(4) = 11 and that simplifies to 11 = 11. In other words, the expression 3(-3)+2(2)+4(4)  turns into 11. This confirms the first equation.
  • Repeat the same idea for the second equation. We have 2x-y+3z=4 turn into 2(-3)-2+3(4) = 4 and that simplifies to 4 = 4. The answer is verified here.
  • The third equation 5x-3y+5z = -1 becomes 5(-3)-3(2)+5(4) = -1 which simplifies to -1 = -1. We get a true equation here as well.

Since all three equations are true when (x, y, z) = (-3, 2, 4), this verifies the answer completely. This is the only solution to the system. The system is independent and consistent.

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