solve the rational equation. 6-x/4-x=3/5
The Correct Answer and Explanation is :
To solve the rational equation
[
\frac{6 – x}{4 – x} = \frac{3}{5},
]
we start by cross-multiplying to eliminate the fractions. This means we multiply both sides by (5(4 – x)):
[
5(6 – x) = 3(4 – x).
]
Expanding both sides gives:
[
30 – 5x = 12 – 3x.
]
Next, we want to isolate (x). First, we can move all terms involving (x) to one side and constant terms to the other side. We can do this by adding (5x) to both sides and subtracting (12) from both sides:
[
30 – 12 = 5x – 3x.
]
This simplifies to:
[
18 = 2x.
]
Now, divide both sides by (2):
[
x = 9.
]
Now we should check our solution by substituting (x = 9) back into the original equation to ensure both sides are equal:
Substituting into the left side:
[
\frac{6 – 9}{4 – 9} = \frac{-3}{-5} = \frac{3}{5}.
]
The right side is:
[
\frac{3}{5}.
]
Since both sides are equal, (x = 9) is indeed a solution.
In summary, we solved the equation by cross-multiplying to eliminate the fractions, simplifying the resulting linear equation, and isolating (x). We found (x = 9), and verified the solution by substituting it back into the original equation to confirm the equality holds. This process is common in solving rational equations, as it allows us to deal with the fractions more easily and ultimately find the value of the variable in question.