How is solving a rational equation different
from solving linear equations like 3x+5=7x-2?
Solving a rational equation is slightly different from solving a linear equation like the one you mentioned, 3x+5=7x-2. While linear equations involve variables raised to the first power (2x, 3y, etc.), rational equations involve variables in the numerator or denominator of a fraction.
The key difference in solving rational equations is the presence of fractions. To solve a rational equation, you typically need to find a common denominator to eliminate the fractions.
Let's consider an example of a rational equation:
1/(x+3) + 1/(x-2) = 2/(x-1)
To solve this type of equation, we follow these steps:
1. Find a common denominator: Multiply each term in the equation by the common denominator of the fractions involved. In this case, the common denominator is (x+3)(x-2)(x-1).
(x-2)(x-1) + (x+3)(x-1) = 2(x+3)(x-2)
2. Simplify: Distribute and combine like terms on each side of the equation.
(x^2 - 3x + 2) + (x^2 + 2x - 3) = 2(x^2 - x - 6)
2x^2 - x - 1 = 2x^2 - 2x - 12
3. Move all terms to one side of the equation: Manipulate the equation to bring all the terms to one side, so that the equation becomes 0.
2x^2 - x - 1 - (2x^2 - 2x - 12) = 0
2x^2 - x - 1 - 2x^2 + 2x + 12 = 0
-x + 11 = 0
4. Solve for x: Solve the resulting linear equation as you would normally do for any linear equation.
-x = -11
x = 11
So, the solution to the given rational equation is x = 11.
In summary, solving rational equations differs from solving linear equations as it involves finding a common denominator to eliminate fractions. The remaining steps, such as simplifying and isolating the variable, are similar to solving linear equations.