We are solving the rational equation x over (x minus 5) equals 2. This is a rational equation because it has a variable in the denominator. Our strategy is to clear the fraction by multiplying both sides by the denominator, then solve the resulting linear equation. We must also check for extraneous solutions that make the denominator zero.
To clear the fraction, we multiply both sides of the equation by (x minus 5). On the left side, the (x minus 5) terms cancel out, leaving just x. On the right side, we distribute the 2 to get 2 times (x minus 5). This simplifies to x equals 2 times (x minus 5).
Now we solve the linear equation. First, distribute the 2 on the right side to get x equals 2x minus 10. Next, subtract 2x from both sides to collect like terms: x minus 2x equals negative 10. This simplifies to negative x equals negative 10. Finally, divide both sides by negative 1 to get x equals 10.
We must verify our solution by substituting x equals 10 back into the original equation. We get 10 over (10 minus 5), which is 10 over 5, equaling 2. Since 2 equals 2 is true, our solution is valid. We also check that our solution doesn't make the denominator zero. Since 10 is not equal to 5, there's no division by zero, confirming our solution is valid.
Let's consider why x equals 5 would be invalid. If we tried to substitute x equals 5 into the original equation, we'd get 5 over (5 minus 5), which is 5 over 0. Division by zero is undefined, so x equals 5 is not in the domain of our function. This is shown graphically by a vertical asymptote at x equals 5. Our solution x equals 10 is valid because it's not equal to 5, and we've verified it satisfies the equation.
We can also solve this equation using cross-multiplication. First, we rewrite 2 as 2 over 1. Then we cross-multiply: x times 1 equals 2 times (x minus 5). This gives us x equals 2 times (x minus 5), which is the same equation we got by multiplying both sides by (x minus 5). From here, we solve as before to get x equals 10. Both methods are equivalent and lead to the same solution.