Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{6}{(x-3)(x-1)}=\frac{3}{x-3}+\frac{4}{x-1} $$

Short answer

The given equation has no solution.

Step-by-step solution

Check Initial Condition

Begin by ensuring x ≠ 1 or 3, since we want to avoid division by zero in this equation.

Clear the Fractions

We need to get rid of the fractions, which we'll do by multiplying both sides of the equation by \((x-3)(x-1)\). Hence we will have \(6 = 3(x - 1) + 4(x - 3)\).

Simplify the Equation

Expand and simplify the expressions on the right side. We obtain \(6 = 3x -3 + 4x -12\), which simplifies further to \(6 = 7x -15\).

Solve for x

Finally, re-arrange the equation to isolate x by doing the following operations; add 15 to both sides, then divide through by 7. This gives us \(x = \frac{21}{7} = 3\).

Check the Solution

Checking our found solution, it appears that \(x = 3\) is not valid, because it makes the original equation undefined (we divide by zero). Therefore, it seems that the equation has no solution.

Key concepts

Rational Equations

Rational equations are equations that involve fractions with polynomials in the numerator, the denominator, or both. In such equations, a variable appears in the denominator, which means we need to be extra cautious about division by zero. Solving rational equations often requires eliminating these fractions to simplify the equation.

• Identify restrictions by determining the values that make the denominator zero.
• Clear the fractions by multiplying each term by the least common denominator (LCD).
• After eliminating the fractions, solve the resulting polynomial equation.
• Finally, test each solution against the original equation to ensure there is no division by zero.

It's essential to keep these steps in mind to solve rational equations effectively while avoiding potential pitfalls such as invalid solutions.

Division by Zero

Division by zero is a critical consideration when dealing with fractional expressions. It occurs when the denominator of a fraction equals zero, something that results in an undefined value in mathematics. To prevent such cases:

• Identify all values of the variable that can lead the denominator to zero.
• State these values as restrictions before solving the equation.
• Pay special attention when checking solutions: ensure they don't violate these restrictions.

In our exercise, we observed that division by zero can invalidate a solution. For the given equation, when substituting values back into the equation, make sure the original terms in the denominator don’t become zero, else that solution is not valid.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions to a simpler form, making solving equations more manageable. You simplify expressions by following these steps:

• Remove parentheses by distributing any outside factors across terms within them.
• Combine like terms, which usually means adding or subtracting terms that share the same variable and exponent.
• Factor expressions to identify common terms for easier manipulation of equations.

In solving rational equations, simplifying expressions helps in both the elimination of fractions and solving the remaining polynomial equation. Simplifying ensures that you're working with the simplest form of the equation, reducing errors and making the problem easier to analyze and solve.