helo sir/madam
can u help me to solve this problem
solve the following simultaneous equation
logx Y+logy X=5/2
xy=64
I saw this one pass by me before.
Are you distinguishing between "x" and "X"?
If so, you have 4 variables with only 2 equations.
I will assume they are the same x
recall that loga b = log b/log a
so logxy = logy/logx
and logyx = logx/logy
so your equation can be written as:
logy/logx + logx/logy = 5/2
let logx/logy = a
then we have
1/a + a = 5/2
times 2a
2 + 2a^2 = 5a
2a^2 - 5a + 2 = 0
(a - 2)(2a - 1) = 0
a = 2 or a = 1/2
if a = 2
logx/logy = 2
logyx = 2
y = x^2
then in xy=64
x(x^2) = 64
x^3 = 64
x = 4 ------------> so x = 4 , y = 16
if a = 1/2
logyx = 1/2
x = y^(1/2) = √y
in xy=64
.....
x = 16, y = 4
Of course, I can help you solve the simultaneous equations.
Let's start by solving the first equation: log(x) y + log(y) x = 5/2.
To simplify this equation, we can use the fact that log(a) b = log(b) a. Using this property, we can rewrite the equation as log(x) y + log(x) y = 5/2.
Combining the logarithms, we get 2 log(x) y = 5/2.
Next, we can rewrite this equation using logarithm rules. Remember that log(a) b + log(a) c = log(a) (b * c). Applying this rule, we get log(x) (y^2) = 5/2.
To eliminate the logarithm, we can exponentiate both sides of the equation. This will give us x^(5/2) = y^2.
Now, let's move on to the second equation: xy = 64.
Based on this equation, we have x * y = 64.
Now that we have simplified both equations, we can solve them simultaneously.
Substituting x^(5/2) from the first equation into the second equation, we have (x^(5/2)) * y = 64.
Raising both sides to the power of 2/5 to eliminate the exponent, we get (x^(5/2))^(2/5) * y^(2/5) = 64^(2/5).
Simplifying further, we have x^2 * y^(2/5) = 64^(2/5).
Now, substitute 64 = 2^6 into the equation, we have x^2 * y^(2/5) = (2^6)^(2/5).
Applying the exponent rule, (a^m)^n = a^(m * n), we can simplify further to x^2 * y^(2/5) = 2^(6 * 2/5).
Performing the calculations, we have x^2 * y^(2/5) = 2^(12/5).
Now, let's simplify the right side: 2^(12/5) = (2^2)^(6/5) = 4^(6/5).
So the equation becomes x^2 * y^(2/5) = 4^(6/5).
To eliminate the exponents, let's raise both sides to the power of 5/2: (x^2 * y^(2/5))^(5/2) = 4^(6/5)^(5/2).
Simplifying, we get x^5 * y = 4^3.
At this point, we have two equations:
1) x^(5/2) = y^2
2) x^5 * y = 64
We can substitute the value of y^2 from the first equation into the second equation:
x^5 * (x^(5/2)) = 64.
To combine the x-terms, we multiply the exponents: x^(15/2) = 64.
Next, we can rewrite 64 as 2^6: x^(15/2) = (2^6).
Simplifying, we have x^(15/2) = 2^6.
To eliminate the fraction exponent, we raise both sides to the power of 2/15: (x^(15/2))^(2/15) = (2^6)^(2/15).
This simplifies to x = 2^(12/15).
Now, we can find the value of y by substituting x = 2^(12/15) into the first equation: (2^(12/15))^(5/2) = y^2.
Simplifying, we have 2^(10/3) = y^2.
Taking the square root of both sides, we get y = ± (2^(10/3))^0.5.
So the solution to the simultaneous equations is x = 2^(12/15) and y = ± (2^(10/3))^0.5.
Please note that the solutions may be further simplified based on specific requirements.