Find all ordered pairs of real numbers (x, y)that satisfy the system of equations log(xy^3) = 5 and log(x^2/y)=3

Question

This would be really helpful to better understand... I would really love if I could get help with this. Thank you!

Answer 1

log (x²/y) = 3 log(xy³) = 5

x²/y = 10³ xy³ = 10⁵

substitute x²/10³ in for y

x² = 10³y x (x²/10³)³ = 10⁵

y = x²/10³ x⁷/10⁹ = 10⁵

x⁷ = 10¹⁴

x = 10² = 100

y = x²/10³ = 100²/10³ = 10

Final Answer (100,10)

Answer 2

Hello Grace,

Since you have two types of variables x's and y's, it's harder to solve systems of equations at a glance. The trick to solving systems like this is to try to get rid of one type of variable. We can't just erase them though, so we need to substitute out something that equals the same thing.

Since we have two equations, we can use one of them to find a relationship between the two variables (in other words, solve for one of them). Then, we use that relationship to replace the variable we solved for, in the second equation.

For example, using the first formula:

log(xy^3) = 5

xy^3 = 10^5

x = 10^5 / y^

Now we know what x is equal to, so we can substitute into the second equation.

log (x^2/y) = 3 becomes

log[(10^5/y^3)/y) = 3

log[(10^10/y^6)/y] = 3

log(10^10/y^7) = 3

10^10/ y^7 = 10^3

y^7 = 10^10/10^3

y^7 = 10^7

y=10

Since we know what y is, we can go back to our equation for x

x = 10^5 / y^3 becomes

x = 10^5 / 10^3

x = 10^2 = 100

Since they're asking for an ordered pair, the answer would be (100,10). (x,y)

Note that it does not matter which variable you solve for first, or which equation you use first. In the end, you'll get the same answer regardless. I could have solved the second equation for y, and then used that to solve for x in the first equation. What's important is to use both equations.

In short:

Step 1. Use one equation to solve for one variable

Step 2. Substitute that variable into the other equation

Step 3. Solve for the second variable

Step 4. Substitute that answer into the equation you found in step 1

I know this is really messy because of the fractions, but I hope you can see the general process. I also glossed over the rules for logarithms, which are needed for this specific question.

Best,

Sebastian