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Solve the system.

x - 3y - 5z = 7
6y + z = 27
6x + 5y - 6z = 85
Enter your answer as an ordered triple.
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1 Answer

2 votes

Answer:

(-6, 4/3, 7)

Explanation:

To solve the system of equations, we can use elimination or substitution. Here, we will use elimination:

First, we can eliminate z by adding the first and third equations:

x - 3y - 5z = 7

6x + 5y - 6z = 85

7x + 2y = 92

Next, we can use the second equation to eliminate z from the second equation:

6y + z = 27

-6z from both sides:

6y = -6z + 27

Divide both sides by 6:

y = -z/6 + 9/2

Now, we can substitute y = -z/6 + 9/2 into the equation 7x + 2y = 92:

7x + 2(-z/6 + 9/2) = 92

7x - z/3 + 9 = 92

7x - z/3 = 83

Multiplying both sides by 3 to eliminate the fraction:

21x - z = 249

We now have two equations:

7x + 2y = 92

21x - z = 249

To eliminate x, we can multiply the first equation by 3 and subtract it from the second equation:

21x - z = 249

- (3 * (7x + 2y = 92))

-----------------------------------------------------------------

-13x - 6z = -215

Solving this equation for z, we get:

-13x - 6z = -215

6z = 13x + 215

z = (13/6)x + 35 + 1/3

Now we can substitute this value of z into one of the equations to solve for y. Let's use the second equation:

6y + z = 27

6y + [(13/6)x + 35 + 1/3] = 27

6y + (13/6)x + 35 + 1/3 - 27 = 0

6y + (13/6)x + 8 + 1/3 = 0

6y + (13/6)x = -25/3

y = (-13/36)x - (25/18)

Finally, we can substitute these values for x, y, and z into one of the original equations to check our work. Let's use the first equation:

x - 3y - 5z = 7

x - 3[(-13/36)x - (25/18)] - 5[(13/6)x + 35 + 1/3] = 7

x + (13/12)x + (25/6) - (65/6)x - 175 - (5/3) = 7

-51x/12 - 215/6 = 7

-17x/4 - 35/2 = 7

-17x/4 = 21/2

x = -6

Substituting this value of x into the expressions we found for y and z, we get:

y = (-13/36)x - (25/18) = (13/6) - (25/18) = 4/3

z = (13/6)x + 35 + 1/3 = -(13/6)*6 + 35 + 1/3 = 7

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