%5Ex-3-(2%2F7)%5Ex-2%3D10%2F49.jpeg "(2/7)^x-3-(2/7)^x-2=10/49")
Solving the Equation (2/7)^x-3 - (2/7)^x-2 = 10/49
This article will guide you through solving the equation (2/7)^x-3 - (2/7)^x-2 = 10/49. We'll break down the steps and provide explanations to make the process clear.
Understanding the Equation
The equation involves exponents with a fractional base (2/7) and variables in the exponent. Our goal is to find the value of x that satisfies the equation.
Simplifying the Equation
We can simplify the equation by factoring out a common term:
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Step 1: Notice that both terms on the left-hand side share the term (2/7)^x-3. Let's factor it out: (2/7)^x-3 * [1 - (2/7)] = 10/49
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Step 2: Simplify the expression inside the brackets: (2/7)^x-3 * (5/7) = 10/49
Isolating the Variable
Now we want to isolate the term (2/7)^x-3:
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Step 3: Divide both sides of the equation by (5/7): (2/7)^x-3 = (10/49) / (5/7)
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Step 4: Simplify the right-hand side: (2/7)^x-3 = 2/7
Solving for x
We can now express both sides of the equation with the same base:
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Step 5: Recognize that 2/7 can be written as (2/7)^1: (2/7)^x-3 = (2/7)^1
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Step 6: Since the bases are the same, we can equate the exponents: x - 3 = 1
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Step 7: Solve for x: x = 4
Solution
Therefore, the solution to the equation (2/7)^x-3 - (2/7)^x-2 = 10/49 is x = 4.
Verification
We can verify our solution by substituting x = 4 back into the original equation:
(2/7)^(4-3) - (2/7)^(4-2) = 10/49 (2/7)^1 - (2/7)^2 = 10/49 2/7 - 4/49 = 10/49
The equation holds true, confirming that x = 4 is the correct solution.