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 rewrite the first equation as 2 = log2(3x+y) and the second equation as -2 = log2(x/y). Using the properties of logarithms, we can rewrite the second equation as 2 = log2(y/x).
Now we can set up a system of equations:
2 = log2(3x+y)
2 = log2(y/x)
Using the inverse property of logarithms, we can rewrite the first equation as 2^2 = 3x+y, or 4 = 3x+y.
Using the change of base formula, we can rewrite the second equation as log2(y) - log2(x) = log2(y/x) = 2.
We can then use substitution to solve for y:
log2(y) - log2((4-y)/3) = 2
log2(y) - log2(4-y) + log2(3) = 2
log2(y(4-y)) = log2(12)
y(4-y) = 12
4y - y^2 = 12
y^2 -4y + 12 = 0
Using the quadratic formula, we get y = 2 +/- 2iโ2
Therefore, there are no real solutions for y.
If we go back to the equation 4 = 3x+y, we can solve for x:
4 = 3x+y
4 = 3x + (2 + 2iโ2)
2 = 3x + iโ8
2/3 = x + iโ8/3
Therefore, x = 2/3 and y = 2 + 2iโ2.
So the answer is none of the above (E).
To find the values of x and y, we will solve the given equations step-by-step.
Step 1: Use the first equation, log2(3๐ฅ + ๐ฆ) = 1, to eliminate the logarithm.
Rewrite the equation in exponential form:
2^1 = 3๐ฅ + ๐ฆ
2 = 3๐ฅ + ๐ฆ
Step 2: Use the second equation, log2 (๐ฅ๐ฆ) = โ2, to eliminate the logarithm.
Rewrite the equation in exponential form:
2^(-2) = ๐ฅ๐ฆ
1/4 = ๐ฅ๐ฆ
Step 3: Now we have the system of equations:
2 = 3๐ฅ + ๐ฆ
1/4 = ๐ฅ๐ฆ
To solve this system of equations, we can use the method of substitution.
Step 4: Solve the second equation, 1/4 = ๐ฅ๐ฆ, for ๐ฅ or ๐ฆ.
Let's solve for ๐ฅ:
1/4 = ๐ฅ๐ฆ
1/4๐ฆ = ๐ฅ
Step 5: Substitute the value of ๐ฅ in the first equation, 2 = 3๐ฅ + ๐ฆ.
2 = 3(1/4๐ฆ) + ๐ฆ
2 = 3/4๐ฆ + ๐ฆ
Step 6: Combine like terms:
Multiply the first term by 4/4 to get a common denominator:
2 = (3/4๐ฆ) * 4/4 + ๐ฆ
2 = 12/4๐ฆ + ๐ฆ
2 = (12 + 4๐ฆ^2)/4๐ฆ
8๐ฆ = 12 + 4๐ฆ^2
Step 7: Rearrange the equation to form a quadratic equation:
4๐ฆ^2 - 8๐ฆ + 12 = 0
Step 8: Simplify the equation by dividing it by 4:
๐ฆ^2 - 2๐ฆ + 3 = 0
Step 9: Solve the quadratic equation using the quadratic formula:
๐ฆ = (-(-2) ยฑ โ((-2)^2 - 4(1)(3))) / (2(1))
๐ฆ = (2 ยฑ โ(4 - 12)) / 2
๐ฆ = (2 ยฑ โ(-8)) / 2
Since the value inside the square root is negative, there are no real solutions for ๐ฆ. Therefore, none of the given options (A, B, C, D) is correct.
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