solve the following equation.
9^(x-1)+4*3^(x-1)=5
120-42÷ (10*4-4*6)
To solve the equation 9^(x-1) + 4*3^(x-1) = 5, we can apply algebraic techniques to isolate x and find its value. Here's how to do it step by step:
Step 1: Simplify the equation.
Start by simplifying the exponential terms using the rules of exponents:
9^(x-1) = (3^2)^(x-1) = 3^(2(x-1)) = 3^(2x-2)
3^(x-1) remains as it is.
The equation now becomes:
3^(2x-2) + 4*3^(x-1) = 5
Step 2: Combine like terms.
We can combine the two terms with a common base of 3:
3^(2x-2) + 4*3^(x-1) = 5
=> 3^(2x-2) + 4*3^(x-1) - 5 = 0
Step 3: Rewrite the equation in terms of a single variable.
Let's introduce a new variable to replace 3^(x-1). Let u = 3^(x-1). This means that u is equal to the base 3 raised to the power of (x-1).
With this substitution, the equation becomes:
u^2 + 4u - 5 = 0
Step 4: Solve the quadratic equation.
Now, we have a quadratic equation we can solve. The equation factors as:
(u + 5)(u - 1) = 0
Setting each factor equal to zero, we find:
u + 5 = 0 => u = -5
u - 1 = 0 => u = 1
Step 5: Relate the variable u back to x.
We substitute back the original expression for u:
u = 3^(x-1)
Setting each solution for u in Step 4 equal to the expression for u, we solve for x:
For u = -5:
3^(x-1) = -5
For u = 1:
3^(x-1) = 1
Step 6: Solve each equation for x.
Solving the equation 3^(x-1) = -5, we recognize that there are no real solutions. The exponential function 3^(x-1) is always positive, so it can never equal -5.
Solving the equation 3^(x-1) = 1, we can take the logarithm base 3 on both sides to simplify:
log3(3^(x-1)) = log3(1)
x-1 = log3(1)
x-1 = 0
x = 1
Step 7: Check the solution.
To verify our solution, substitute x = 1 back into the original equation:
9^(1-1) + 4*3^(1-1) = 5
1 + 4*1 = 5
1 + 4 = 5
5 = 5
The equation holds true, so x = 1 is the solution.