Solve this mathematically?

Question

49^x + 7^1-2x = 8

Do NOT give me just the answer, I already know what they are. Just tell me how to do it mathematically. Thank you : )

Answer 1

I think you made a mistake when you wrote the equation and maybe that is why you have no answer yet.

I suppose you want to solve the equation 49^x + 7^(1-2x) = 8, which is something different.

So let us solve it step by step:

49^x + 7^(1-2x) = 8
49^x + 1/7^(2x - 1) = 8
49^x + 7/7^(2x) = 8
49^x + 7/49^x = 8
49^(2x) + 7 = 8 * 49^x
49^(2x) - 8 * 49^x + 7 = 0

To make this equation more transparent let us introduce a substitution y = 49^x.

y^2 – 8y + 7 = 0

This quadratic equation has two solutions

y = 7 and y = 1

Now let us return to solve x.

For y = 1 we get

1 = 49^x
x = 0

For y = 7 we get

7 = 49^x
x = 0.5

So the answer is

x = 0
and
x = 0.5

If some of the steps is not quite clear ask again.

-