Question

X^2+y^2=5. log(a)x+log(a)y=log(a)2​

Answer 1

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Answer 2

To solve the system of equations:

1. x^2 + y^2 = 5
2. log(a)x + log(a)y = log(a)2

Let's start by addressing the second equation. The logarithm property that will be useful in this case is:

log(a)mn = log(a)m + log(a)n

Using this property, we can rewrite the second equation as:

log(a)(xy) = log(a)2

Now, since both sides of the equation have the same base (logarithm base 'a'), we can drop the logarithm notation and write:

xy = 2

Now we have two equations:

1. x^2 + y^2 = 5
2. xy = 2

We can use substitution or elimination to solve this system. Let's use substitution. Solve equation 2 for x or y and substitute the value into equation 1.

From equation 2, we can express x in terms of y:

x = 2/y

Now substitute this value of x into equation 1:

(2/y)^2 + y^2 = 5

Simplify the equation:

4/y^2 + y^2 = 5

Multiply both sides by y^2 to get rid of the denominators:

4 + y^4 = 5y^2

Rearrange the equation to set it to zero:

y^4 - 5y^2 + 4 = 0

This is now a quadratic equation in terms of y^2. We can solve for y^2 using the quadratic formula:

y^2 = [5 ± sqrt(5^2 - 414)] / 2 y^2 = [5 ± sqrt(9)] / 2 y^2 = (5 ± 3) / 2

This gives two possible values for y^2:

1. y^2 = (5 + 3) / 2 = 4
2. y^2 = (5 - 3) / 2 = 1

Now, find the corresponding values of y by taking the square root of each y^2 value:

1. y = sqrt(4) = 2
2. y = sqrt(1) = 1

Now that we have two possible values for y (y = 2 and y = 1), we can find the corresponding values of x using equation 2:

For y = 2: x = 2/y = 2/2 = 1

For y = 1: x = 2/y = 2/1 = 2

So, the solutions to the system of equations are (x = 1, y = 2) and (x = 2, y = 1).