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If (3,-1,2) is a solution of the following system

x+ay+cz=0, bx+cy-3z=1, and ax+2y+bz=25, then a+b+c=

User Jordash
by
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1 Answer

4 votes

Answer:

I'm not so sure

Explanation:

Let's solve the system of equations again using the given solution.

1) x + ay + cz = 0

Substituting x = 3, y = -1, z = 2:

3 + a(-1) + c(2) = 0

3 - a + 2c = 0

- a + 2c = -3 ---(Equation 1)

2) bx + cy - 3z = 1

Substituting x = 3, y = -1, z = 2:

b(3) + c(-1) - 3(2) = 1

3b - c - 6 = 1

3b - c = 7 ---(Equation 2)

3) ax + 2y + bz = 25

Substituting x = 3, y = -1, z = 2:

a(3) + 2(-1) + b(2) = 25

3a - 2 + 2b = 25

3a + 2b = 27 ---(Equation 3)

To solve this system, we can use the method of substitution.

From Equation 1, we can express a in terms of c:

- a + 2c = -3

a = 2c - 3

Substitute this value of a in Equation 3:

3a + 2b = 27

3(2c - 3) + 2b = 27

6c - 9 + 2b = 27

6c + 2b = 36 ---(Equation 4)

Now we have two equations with two variables, Equation 2 and Equation 4:

3b - c = 7 ---(Equation 2)

6c + 2b = 36 ---(Equation 4)

We can solve this system of equations to find the values of b and c.

Multiplying Equation 2 by 2, we get:

6b - 2c = 14

Adding this equation to Equation 4, we eliminate c:

6c + 2b + 6b - 2c = 36 + 14

8b = 50

b = 50/8

b = 6.25

Substituting the value of b back into Equation 2, we can solve for c:

3(6.25) - c = 7

18.75 - c = 7

-c = 7 - 18.75

-c = -11.75

c = 11.75

Now that we have the values of b and c, we can substitute them into Equation 1 to solve for a:

- a + 2(11.75) = -3

- a + 23.5 = -3

- a = -3 - 23.5

- a = -26.5

a = 26.5

Finally, we can calculate the value of a + b + c:

a + b + c = 26.5 + 6.25 + 11.75

a + b + c = 44.5

Therefore, a + b + c is equal to 44.5.

User Daniel Romero
by
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