Consider the following eqution
x2 + 4y2 + 9z2 = 14x + 28y+42z + 147
Then the value of x+2y+3z is
21
To find the value of x + 2y + 3z, we need to rearrange the given equation and isolate the terms involving x, y, and z.
Rearranging the equation, we have:
x^2 + 4y^2 + 9z^2 - 14x - 28y - 42z = 147
Now, we can complete the square for each variable.
For x:
(x^2 - 14x) + 4y^2 + 9z^2 - 28y - 42z = 147
Completing the square, we add (14/2)^2 = 49 to both sides:
(x^2 - 14x + 49) + 4y^2 + 9z^2 - 28y - 42z = 147 + 49
(x - 7)^2 + 4y^2 + 9z^2 - 28y - 42z = 196
For y:
(x - 7)^2 + (4y^2 - 28y) + 9z^2 - 42z = 196
Completing the square, we add (28/2)^2 = 196 to both sides:
(x - 7)^2 + (4y^2 - 28y + 196) + 9z^2 - 42z = 196 + 196
(x - 7)^2 + (2y - 14)^2 + 9z^2 - 42z = 392
For z:
(x - 7)^2 + (2y - 14)^2 + (9z^2 - 42z) = 392
Completing the square, we add (42/2)^2 = 441 to both sides:
(x - 7)^2 + (2y - 14)^2 + (9z^2 - 42z + 441) = 392 + 441
(x - 7)^2 + (2y - 14)^2 + (3z - 21)^2 = 833
Now, we can rewrite the equation as:
(x - 7)^2 + (2y - 14)^2 + (3z - 21)^2 = 833
Comparing this to the standard equation of a sphere:
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2
We can see that the center of the sphere is at (a, b, c) = (7, 7, 21), and the radius, r, is √833.
Now we can find the value of x + 2y + 3z:
x + 2y + 3z = (x - 7) + 2(y - 7) + 3(z - 21)
= (x - 7) + 2y - 14 + 3z - 63
= x + 2y + 3z - 7 - 14 - 63
= x + 2y + 3z - 84
Therefore, the value of x + 2y + 3z is 84.
To find the value of x + 2y + 3z, we need to manipulate the given equation and solve for x, y, and z individually.
First, let's rewrite the equation in standard form by grouping the x, y, and z terms on one side and moving all constant terms to the other side:
x^2 + 4y^2 + 9z^2 - 14x - 28y - 42z = 147
Next, let's complete the square for each variable individually.
For x:
Rearrange the x terms to group them:
x^2 - 14x = 4y^2 + 9z^2 - 28y - 42z + 147
To complete the square, we need to divide the coefficient of x by 2 and square it. Half of -14 is -7, and (-7)^2 is 49. Add 49 to both sides of the equation:
x^2 - 14x + 49 = 4y^2 + 9z^2 - 28y - 42z + 147 + 49
Simplify:
(x - 7)^2 = 4y^2 + 9z^2 - 28y - 42z + 196
Repeat the same steps for y and z:
For y:
4y^2 - 28y = 9z^2 - 42z + 196 - 147
Divide the coefficient of y by 2 and square it. Half of -28 is -14, and (-14)^2 is 196. Add 196 to both sides:
4y^2 - 28y + 196 = 9z^2 - 42z + 196 - 147
Simplify:
4(y - 7)^2 = 9z^2 - 42z + 49
For z:
9z^2 - 42z = 4(y - 7)^2 - 49
Divide the coefficient of z by 2 and square it:
(9/4)z^2 - (42/4)z = 4(y - 7)^2 - 49
Simplify:
(9/4)(z^2 - 2z) = 4(y - 7)^2 - 49
Now that we have completed the squares, let's rearrange the equation to solve for x, y, and z separately:
1. (x - 7)^2 = 4y^2 + 9z^2 - 28y - 42z + 196
2. 4(y - 7)^2 = 9z^2 - 42z + 49
3. (9/4)(z^2 - 2z) = 4(y - 7)^2 - 49
Now, we can solve for x, y, and z using these equations. Once we find their values, we can substitute them back into x + 2y + 3z and calculate the result.