Okay,
How would you solve an equation that has 3 or 4 variables and is set up to something like this:
3^(x+5) times 4^(2y) times 6^(z+8) = 3^(y+9) times 4^(3z) times 6^(4x)
And for the Geometric Series:
"The sum of an infinite geometric series with the first term "a" and a common ratio of "r < 1" is given by "a / 1-r". The sum of a given infinite geometric series is 200, and the common ratio is 0.15. What is the second term of this series?
F. 25.5
G. 30
H. 169.85
J. 170
K. 199.85
Please show as much understandable work as possible, and I'll try to get the best answer for the 10 points in there! Thanks!
How would you solve an equation that has 3 or 4 variables and is set up to something like this:
3^(x+5) times 4^(2y) times 6^(z+8) = 3^(y+9) times 4^(3z) times 6^(4x)
And for the Geometric Series:
"The sum of an infinite geometric series with the first term "a" and a common ratio of "r < 1" is given by "a / 1-r". The sum of a given infinite geometric series is 200, and the common ratio is 0.15. What is the second term of this series?
F. 25.5
G. 30
H. 169.85
J. 170
K. 199.85
Please show as much understandable work as possible, and I'll try to get the best answer for the 10 points in there! Thanks!
-
3^(x + 5) * 4^(2y) * 6^(z + 8) = 3^(y + 9) * 4^(3z) * 6^(4x)
3^(x + 5) * (2^2)^(2y) * (2 * 3)^(z + 8) = 3^(y + 9) * (2^2)^(3z) * (2 * 3)^(4x)
3^(x + 5) * 3^(z + 8) * 2^(4y) * 2^(z + 8) = 3^(y + 9) * 2^(6z) * 2^(4x) * 3^(4x)
3^(x + 5 + z + 8) * 2^(4y + z + 8) = 3^(y + 9 + 4x) * 2^(6z + 4x)
3^(x + z + 13) * 2^(4y + z + 8) = 3^(y + 4x + 9) * 2^(6z + 4x)
3^(x + z + 13) = 3^(y + 4x + 9)
x + z + 13 = y + 4x + 9
13 - 9 = 4x - x + y - z
4 = 3x + y - z
2^(4y+ z + 8) = 2^(6z + 4x)
4y + z + 8 = 6z + 4x
8 = 4x - 4y + 6z - z
8 = 4x - 4y + 5z
z = 3x + y - 4
8 = 4x - 4y + 5 * (3x + y - 4)
8 = 4x - 4y + 15x + 5y - 20
28 = 19x + y
y = 28 - 19x
z = 3x + y - 4
z = 3x + 28 - 19x - 4
z = 24 - 16x
Since there are really only 2 equations and 3 variables, then the best we can do is solve for a variable in terms of another variable
x = x
y = 28 - 19x
z = 24 - 16x
S = a / (1 - r)
200 = a / (1 - 0.15)
200 = a / 0.85
200 * 0.85 = a
170 = a
a * r =>
170 * 0.15 =>
25.5
3^(x + 5) * (2^2)^(2y) * (2 * 3)^(z + 8) = 3^(y + 9) * (2^2)^(3z) * (2 * 3)^(4x)
3^(x + 5) * 3^(z + 8) * 2^(4y) * 2^(z + 8) = 3^(y + 9) * 2^(6z) * 2^(4x) * 3^(4x)
3^(x + 5 + z + 8) * 2^(4y + z + 8) = 3^(y + 9 + 4x) * 2^(6z + 4x)
3^(x + z + 13) * 2^(4y + z + 8) = 3^(y + 4x + 9) * 2^(6z + 4x)
3^(x + z + 13) = 3^(y + 4x + 9)
x + z + 13 = y + 4x + 9
13 - 9 = 4x - x + y - z
4 = 3x + y - z
2^(4y+ z + 8) = 2^(6z + 4x)
4y + z + 8 = 6z + 4x
8 = 4x - 4y + 6z - z
8 = 4x - 4y + 5z
z = 3x + y - 4
8 = 4x - 4y + 5 * (3x + y - 4)
8 = 4x - 4y + 15x + 5y - 20
28 = 19x + y
y = 28 - 19x
z = 3x + y - 4
z = 3x + 28 - 19x - 4
z = 24 - 16x
Since there are really only 2 equations and 3 variables, then the best we can do is solve for a variable in terms of another variable
x = x
y = 28 - 19x
z = 24 - 16x
S = a / (1 - r)
200 = a / (1 - 0.15)
200 = a / 0.85
200 * 0.85 = a
170 = a
a * r =>
170 * 0.15 =>
25.5