Find the value of the variable that would make each equation true.
1. 7^x^2 = 7^2x-1
2. 9^2(x-1) = 9^3x+1
#1
x^2 = 2x-1
x = 1
#2
2(x-1) = 3x+1
x = -3
To find the value of the variable that would make each equation true, we need to solve the equations.
1. 7^x^2 = 7^2x-1
To solve this equation, we need to equate the exponents. Since both sides of the equation have bases of 7, we can set the exponents equal to each other.
x^2 = 2x - 1
Now, we have a quadratic equation. To solve it, we can rearrange the equation to the standard form:
x^2 - 2x + 1 = 0
This equation can be factored as:
(x - 1)(x - 1) = 0
From the factored form, we see that one solution exists: x = 1.
Therefore, the value of the variable that makes the equation true is x = 1.
2. 9^2(x-1) = 9^3x+1
Similar to the previous equation, we need to equate the exponents because the bases are the same.
2(x - 1) = 3x + 1
Now, we can solve for x.
2x - 2 = 3x + 1
Rearranging the equation:
2x - 3x = 1 + 2
-x = 3
Dividing both sides of the equation by -1 to solve for x:
x = -3
Therefore, the value of the variable that makes the equation true is x = -3.