Algebraic Expression Evaluation
In this article, we will evaluate the algebraic expression:
1/(x-1) + 1/(x-2) - 1/(x-3) = 2/3
To evaluate this expression, we need to follow the order of operations (PEMDAS):
Step 1: Simplify the Fractions
Let's start by simplifying each fraction:
- 1/(x-1): This fraction is already in its simplest form.
- 1/(x-2): This fraction is already in its simplest form.
- -1/(x-3): This fraction is already in its simplest form.
Step 2: Combine the Fractions
Now, let's combine the fractions:
- 1/(x-1) + 1/(x-2) = (x-2 + x-1) / ((x-1)(x-2))
- = (2x-3) / (x^2 - 3x + 2)
Subtract -1/(x-3) from the result:
- (2x-3) / (x^2 - 3x + 2) - 1/(x-3)
- = ((2x-3)(x-3) - 1) / ((x-1)(x-2)(x-3))
Step 3: Simplify the Result
Now, let's simplify the resulting fraction:
- ((2x-3)(x-3) - 1) / ((x-1)(x-2)(x-3))
- = (2x^2 - 6x + 3) / (x^3 - 6x^2 + 11x - 6)
Step 4: Equate the Expression
Finally, let's equate the expression to 2/3:
- (2x^2 - 6x + 3) / (x^3 - 6x^2 + 11x - 6) = 2/3
Now, we can solve for x by multiplying both sides by the denominator and simplifying the resulting equation.
By following these steps, we have successfully evaluated the algebraic expression and equated it to 2/3.
