We are solving the rational equation x over (x minus 3) equals 2. This is a rational equation because it has a variable in the denominator. Our goal is to find the value of x that makes this equation true. We'll use a systematic approach and check for any restrictions on x.
In rational equations, the denominator cannot be zero because division by zero is undefined. For our equation, the denominator is x minus 3. We set x minus 3 not equal to zero, which means x cannot equal 3. We'll show this on a number line with an open circle at x equals 3, indicating this value is excluded from the domain.
To solve the equation, we can use cross multiplication. First, we rewrite 2 as 2 over 1. Then we cross multiply: x times 1 equals 2 times (x minus 3). This gives us x equals 2 times (x minus 3).
Now we solve the linear equation x equals 2 times (x minus 3). First, distribute the 2 to get x equals 2x minus 6. Then subtract 2x from both sides to get negative x equals negative 6. Finally, multiply both sides by negative 1 to find x equals 6.
We must verify our solution by substituting x equals 6 back into the original equation. We get 6 over (6 minus 3), which is 6 over 3, equaling 2. Since both sides equal 2, our solution is correct. Also, x equals 6 does not violate our domain restriction since 6 is not equal to 3.
An alternative approach is to multiply both sides of the equation by the denominator (x minus 3). This eliminates the fraction directly, giving us x equals 2 times (x minus 3). We then solve this equation as before to find x equals 6. This method is equivalent to cross multiplication but shows the underlying principle more clearly.
When solving rational equations, students often make several common mistakes. First, they forget to check domain restrictions. Second, they make algebraic errors during distribution or simplification. Third, they don't verify their solution by substituting it back into the original equation. For example, if we mistakenly found x equals 3, substituting it would result in division by zero, which is undefined. Always remember to check your solution.
To summarize our approach to solving rational equations: First, identify domain restrictions by setting denominators not equal to zero. Second, eliminate fractions using cross multiplication or multiplying by the least common denominator. Third, solve the resulting equation using standard algebraic techniques. Fourth, always check your solution by substituting it back into the original equation. Our final answer is x equals 6, which we've verified satisfies both the equation and domain requirements.