The solution of the equation \(\frac{2}{3 x-4}+\frac{2}{2 x-6}=0 \) is:
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Finding the Solution of the Equation
We are asked to find the solution of the equation given by \(\frac{2}{3 x-4}+\frac{2}{2 x-6}=0 \). This is a rational equation because it involves algebraic fractions.
To find the solution of the equation, we need to isolate the variable \(x\). Let's start by rearranging the terms.
Steps to Solve the Rational Equation
Here are the steps to solve for \(x\):
- Rewrite the equation by moving one term to the other side:
\(\frac{2}{3 x-4}=-\frac{2}{2 x-6}\)
- Divide both sides of the equation by 2:
\(\frac{1}{3 x-4}=-\frac{1}{2 x-6}\)
- To eliminate the denominators, we can cross-multiply or multiply both sides by the least common multiple of the denominators, which is \((3x-4)(2x-6)\). This gives:
\(1 \cdot (2x-6) = -1 \cdot (3x-4)\)
- Simplify both sides of the equation:
\(2x - 6 = -3x + 4\)
- Collect terms with \(x\) on one side and constant terms on the other. Add \(3x\) to both sides:
\(2x + 3x - 6 = -3x + 3x + 4\)
\(5x - 6 = 4\)
- Add 6 to both sides:
\(5x - 6 + 6 = 4 + 6\)
\(5x = 10\)
- Divide both sides by 5 to find the solution for \(x\):
\(x = \frac{10}{5}\)
\(x = 2\)
Verifying the Solution
It's important to check if the solution \(x=2\) makes any denominator in the original equation zero. The denominators are \(3x-4\) and \(2x-6\).
- For \(x=2\), \(3x-4 = 3(2) - 4 = 6 - 4 = 2\). This is not zero.
- For \(x=2\), \(2x-6 = 2(2) - 6 = 4 - 6 = -2\). This is not zero.
Since neither denominator is zero when \(x=2\), the solution \(x=2\) is valid for the solution of the equation \(\frac{2}{3 x-4}+\frac{2}{2 x-6}=0 \). The steps taken to solve this algebraic equation correctly lead to this solution.
Thus, the solution of the equation is \(x=2\).
Important Questions from Simplification
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