Solve the problem.
-Three solutions to the equation Ax + By + Cz = 3 are (1, -4, 2), (1, 8, -4), and (1, 6, -3). Determine the values of A, B, and C and write the equation using the numerical values found.
A) A = 3, B = -4, C = -2; 3x - 4y - 2z = 3
B) A = -2, B = 3, C = -4; -2x + 3y - 4z = 3
C) A = 3, B = -2, C = -4; 3x - 2y - 4z = 3
D) A = -4, B = -2, C = 3; -4x - 2y + 3z = 3

Question

Quizplus · Accepted Answer

To find the values of A, B, and C, we can plug in each of the given solutions for x, y, and z and solve for the variables. Using the first point (1, -4, 2):
A(1) + B(-4) + C(2) = 3
A - 4B + 2C = 3
Using the second point (1, 8, -4):
A(1) + B(8) + C(-4) = 3
A + 8B - 4C = 3
Using the third point (1, 6, -3):
A(1) + B(6) + C(-3) = 3
A + 6B - 3C = 3
Now we have a system of three equations with three variables. We can solve for A, B, and C using elimination or substitution:
Multiplying the first equation by 2 and adding it to the third equation:
2A - 8B + 4C = 6
A + 6B - 3C = 3
3A - 2B + C = 3
Multiplying the second equation by 3 and adding it to the third equation:
3A + 24B - 12C = 9
A + 6B - 3C = 3
4A + 22B = 12C + 3
Multiplying the second equation by 2 and adding it to the first equation:
2A - 8B + 4C = 6
A + 8B - 4C = 3
3A = 9
A = 3
Substituting A = 3 into the third equation:
3(3) - 2B + C = 3
9 - 2B + C = 3
-2B + C = -6
2B - C = 6
Adding this equation to the first equation:
2A - 8B + 4C = 6
2B - C = 6
6 - 6B + 5C = 12
-6B + 5C = 6
Multiplying this equation by 2 and adding it to the second equation:
4A + 22B = 12C + 3
-12B + 10C = 12
-6B + 5C = 6
Multiplying this equation by 2 and adding it to the last equation:
-12B + 10C = 12
-12B + 10C = 12
Therefore, B = -2 and C = -4. Plugging these values back into any of the three equations, we get that they are consistent with the original solutions. Thus, the equation is:

3x - 2y - 4z = 3

which matches choice C. We can use the given solutions to form a system of three equations with three variables:
A(1) + B(-4) + C(2) = 3
A(1) + B(8) + C(-4) = 3
A(1) + B(6) + C(-3) = 3

Simplifying each equation:
A - 4B + 2C = 3
A + 8B - 4C = 3
A + 6B - 3C = 3

We can then use elimination to solve for A, B, and C.
Multiplying the first equation by -2 and adding it to the second equation:
-2A + 8B - 4C = -6
A + 8B - 4C = 3
------------------
- A + 0B - 8C = -3

Multiplying the first equation by -1 and adding it to the third equation:
-A + 4B - 2C = -3
A + 6B - 3C = 3
------------------
0A + 10B - 5C = 0

Multiplying the second equation by 2 and adding it to the third equation:
A + 8B - 4C = 3
0A + 10B - 5C = 0
------------------
A + 26B - 13C = 6

Now we have the system:
-A + 0B - 8C = -3
0A + 10B - 5C = 0
A + 26B - 13C = 6

Solving for A, B, and C:
Multiplying the second equation by 2 and adding it to the third equation:
A + 26B - 13C = 6
0A + 20B - 10C = 0
------------------
A + 46B - 23C = 6

Multiplying the first equation by 5 and adding it to the third equation:
-5A + 0B + 40C = -15
A + 46B - 23C = 6
-------------------
A + 46B + 17C = -9

Multiplying the first equation by 46 and adding it to the second equation:
-46A + 0B + 368C = -138
A + 46B + 17C = -9
---------------------
-45A + 46B + 385C = -147

Solving the system:
-A + 0B - 8C = -3
A + 46B + 17C = -9
-45A + 46B + 385C = -147

We get A = 3, B = -2, and C = -4, which matches choice C:
3x - 2y - 4z = 3

Solve the Problem. -Three Solutions to the Equation Ax + by + Cz