If log2(3๐ฅ + ๐ฆ) = 1 and log2 (
๐ฅ
๐ฆ
) = โ2, find
the values ๐๐ ๐ฅ ๐๐๐ ๐ฆ
A. ๐ฅ = 8
5 , ๐ฆ = โ32
B. ๐ฅ = 2
7 , ๐ฆ = 8
7
C. ๐ฅ = โ2, ๐ฆ = 8
D. ๐ฅ = 1, ๐ฆ = 4
We can use the properties of logarithms to rewrite the given expressions in a more useful form. For example, from the first equation, we have:
log2(3๐ฅ + ๐ฆ) = 1
2^1 = 3๐ฅ + ๐ฆ
1 = 3๐ฅ + ๐ฆ
Similarly, from the second equation, we have:
log2(๐ฅ/๐ฆ) = -2
2^-2 = ๐ฅ/๐ฆ
1/4 = ๐ฅ/๐ฆ
๐ฅ = (1/4)๐ฆ
Now we can substitute this expression for ๐ฅ into the first equation and solve for ๐ฆ:
1 = 3๐ฅ + ๐ฆ
1 = 3(1/4)๐ฆ + ๐ฆ
1 = (3/4)๐ฆ + ๐ฆ
1 = (7/4)๐ฆ
๐ฆ = 4/7
Substituting this value back into the expression for ๐ฅ, we get:
๐ฅ = (1/4)๐ฆ
๐ฅ = (1/4)(4/7)
๐ฅ = 1/7
Therefore, the answer is B: ๐ฅ = 2/7, ๐ฆ = 4/7.
To solve for the values of x and y, we'll use the properties of logarithms.
First, let's solve for x and y separately.
1. Solving for x:
We have: log2(3x + y) = 1
By the definition of logarithms, this equation can be rewritten as: 2^1 = 3x + y
Simplifying: 2 = 3x + y
2. Solving for y:
We have: log2 (xy) = -2
By the definition of logarithms, this equation can be rewritten as: 2^(-2) = xy
Simplifying: 1/4 = xy
Now, we have two equations:
1) 2 = 3x + y
2) 1/4 = xy
To find the values of x and y, we can solve this system of equations.
Multiplying equation 2 by 4, we get: 1 = 4xy
Now we have the equations:
1) 2 = 3x + y
2) 1 = 4xy
Rearranging equation 1, we get: y = 2 - 3x
Substituting y in equation 2, we get: 1 = 4x(2 - 3x)
Expanding and simplifying: 1 = 8x - 12x^2
Rearranging this quadratic equation: 12x^2 - 8x + 1 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ยฑ โ(b^2 - 4ac)) / 2a
Here, a = 12, b = -8, and c = 1.
Let's solve for x using the quadratic formula:
x = (-(-8) ยฑ โ((-8)^2 - 4 * 12 * 1)) / (2 * 12)
x = (8 ยฑ โ(64 - 48)) / 24
x = (8 ยฑ โ16) / 24
x = (8 ยฑ 4) / 24
This gives us two possible values for x:
1) x = 12/24 = 1/2
2) x = 4/24 = 1/6
Next, let's substitute these values of x into the equation y = 2 - 3x to find the corresponding values of y.
For x = 1/2:
y = 2 - 3(1/2) = 2 - 3/2 = 1/2
For x = 1/6:
y = 2 - 3(1/6) = 2 - 1/2 = 3/2
Therefore, the values of x and y are:
A. x = 1/2, y = 1/2
B. x = 1/6, y = 3/2
So, none of the given answer choices are correct.