Solve for x and y if 5^(x−y) = 625 and 3^(2x) × 3^y = 243.
(note: I posted a similar question to this before, but this uses indices)
(Also note: Answer is, x = 3, y = -1 ; 7, and the question if inputted into an online calculator, is incorrect.)
Thnks for your time.
since 5^4 = 625
and 3^(2x)*3^y = 3^(2x+y) and 243 = 3^5
x-y = 4
2x+y = 5
not so hard now, eh?
hm. I feel so stupid now.
Well, this is a far cry from what you made me do yesterday after you posted this:
https://www.jiskha.com/display.cgi?id=1511439208
btw, give yourself a nickname instead of "anonymous" to find your posts easier.
yeah, extremely sorry about that, though the question was to compare these two numbers (one w/ and w/o indices), and I only gave you half of the pie, so I'm really sorry about that. But, I'll try to think of a nickname, thanks for the suggestion.
To solve the given equations, we will use the properties of exponents and logarithms.
Let's start with the first equation:
5^(x - y) = 625
Since 625 can be written as 5^4, we can rewrite the equation as:
5^(x - y) = 5^4
Now we can equate the exponents:
x - y = 4
Next, let's move on to the second equation:
3^(2x) * 3^y = 243
Since 243 can be written as 3^5, we can rewrite the equation as:
3^(2x) * 3^y = 3^5
Using the properties of exponents, we can add the exponents on the left side:
3^(2x + y) = 3^5
Now we can equate the exponents:
2x + y = 5
We have a system of two equations:
x - y = 4
2x + y = 5
To solve this system, we can use the method of substitution. Solve one equation for one variable and substitute it into the other equation.
From the first equation, we can solve for x:
x = 4 + y
Now substitute this expression for x into the second equation:
2(4 + y) + y = 5
Simplify the equation:
8 + 2y + y = 5
3y + 8 = 5
3y = 5 - 8
3y = -3
y = -1
Substitute the value of y back into the first equation to find x:
x - (-1) = 4
x + 1 = 4
x = 4 - 1
x = 3
Therefore, the solution to the given system of equations is:
x = 3
y = -1