SOLUTION: Hello I have a question I'm really stuck on. The question says solve this simultaneous equation 2^x - 3^y = -1 2^x-2 - 3^y-1 = -1 Please could you help me solve this. Th

Question 664573: Hello
I have a question I'm really stuck on. The question says solve this simultaneous equation
2^x - 3^y = -1
2^x-2 - 3^y-1 = -1
Please could you help me solve this. Thank you very much in advance.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
clever professor - mixes an exponential problem with a simultaneous equation problem. you have my sympathy

the key to the solution is knowing that x^a / x^b = x^(a-b)
once you know that you're a little less than half way there.

the problem statement is to solve the following equations simnultaneously.

2^x - 3^y = -1
2^x-2 - 3^y-1 = -1

2^x-2 is equivalent to 2^x / 2^2 which is equivalent to 2^x / 4

3^y-1 is equivalent to 3^y/3^1 which is equivalent to 3^y / 3

substitute in your original equations and they become:

2^x - 3^y = - 1
2^x/4 - 3^y/3 = - 1

we will solve this simultaneous by substitution.

solve for 2^x in the first equation to get 2^x = 3^y - 1

substitute for 2^x in the second equation to get ((3^y) - 1)/4 - 3^y/3 = - 1

multiply both sides of the equation by 12 to get:

12 * ((3^y) - 1) / 4 - 12 * (3^y)/3 = 12 * - 1

simplify to get:

3 * ((3^y)-1) - 4 * (3^y) = - 12

simplify further to get:

3 * (3^y) - 3 - 4 * (3^y) = - 12

combine like terms to get:

- (3^y) - 3 = - 12

add 3 to both sides of the equation to get:

- (3^y) = - 9

multiply both sides of this equation by -1 to get 3^y = 9

since 3^2 = 9, y must be equal to 2 **********************

in the equation 2^x = 3^y - 1, replace y with 2.
this makes the equation 2^x = 3^2 - 1 which makes the equation 2^x = 9 - 1 which makes the equation 2^x = 8

since 2^3 = 8, then x must be equal to 3 **********************

you have x = 3 and y = 2 and all that remains is to go back to the original equation and confirm that these solutions are good.

the original equations are:

2^x - 3^y = -1
2^x-2 - 3^y-1 = -1

replace with 3 and y with 2 in these equations to get:

2^3 - 3^2 = -1
2^3-2 - 3^2-1 = -1

simplify these equations to get:

2^3 - 3^2 = -1 becomes 8 - 9 = -1 which becomes -1 = -1 which is true.
2^3-2 - 3^2-1 = -1 becomes 2^1 - 3^1 = -1 which becomes 2 - 3 = -1 which becomes -1 = -1 which is also true.

the values for x and y solve both original equations so they're good.

RELATED QUESTIONS
Hello I'm really stuck on this question and would be very grateful if you could help... (answered by jim_thompson5910)

please I need help on this how to solve this i'm stuck the question read as follow :use... (answered by Fombitz)

What method do i have to use to solve this question? I'm really stuck. Find the... (answered by robertb)

hello i have a question on the equation of an Ellipse i have the equation of... (answered by scott8148)

Hello, Please help me solve this. Stuck on it for 2 days and keep getting different... (answered by Cromlix)

hello, i'm really confused how to start off this equation; i've been told that i have to... (answered by richwmiller)

Solve this simultaneous equation: {{{(3/x) - (4/y) = (1/3)}}}, {{{(2/x) - (5/y)... (answered by ikleyn)

Hello tutor, I really need your help to solve this question please. Solve the... (answered by checkley77,Edwin McCravy)

Solve 9-x^2=y^2, x+y=3, as a linear simultaneous... (answered by KMST)