Question

The solution set of equation \([(\mathrm{x}-\mathrm{a}) /(\mathrm{x}-\mathrm{b})]+\) \([(x-b) /(x-a)]=(a / b)+(b / a),(a \neq b)\) is (a) \(\\{\mathrm{a}-\mathrm{b}, 0\\}\) (b) \(\\{(\mathrm{a} / \mathrm{b}), 0\\}\) (c) \(\\{a+b, 0\\}\) (d) \(\\{\mathrm{ab}, 0\\}\)

Short answer

The short answer is: The solution set of the equation is (c) $\{a+b, 0\}$.

Step-by-step solution

Identify the given equation and the restrictions

The equation to solve is: \[(\frac{x-a}{x-b})+(\frac{x-b}{x-a})=\frac{a}{b}+\frac{b}{a}\] With the restriction: \(a \neq b\).

Find a common denominator for the fractions on the left side.

In order to simplify and solve the equation, we'll need to find a common denominator for the fractions \(x-a\) and \(x-b\). In this case, the common denominator will be \((x-a)(x-b)\).

Combine the two fractions on the left side of the equation using the common denominator.

Now that we have a common denominator, let’s combine the fractions on the left side of the equation: \[\frac{(x-a)(x-b)+(x-b)(x-a)}{(x-a)(x-b)}=\frac{a}{b}+\frac{b}{a}\]

Simplify the numerator of the left side of the equation.

We can simplify the numerator of the left fraction by expanding the terms and then simplifying: \[\frac{(x^2-ax-bx+ab)+(x^2-bx-ax+ab)}{(x-a)(x-b)}=\frac{a}{b}+\frac{b}{a}\] \[\frac{2x^2-2ax-2bx+2ab}{(x-a)(x-b)}=\frac{a}{b}+\frac{b}{a}\]

Cross multiply the left side and right side of the equation.

To eliminate the fractions, we can cross multiply: \[(2x^2-2ax-2bx+2ab)(ab)=(x-a)(x-b)(a+b)\]

Expand and simplify the equation.

First, expand the left side of the equation: \(2x^2ab-2a^2bx-2ab^2x+2a^2b^2=(x-a)(x-b)(a+b)\) Then, expand the right side of the equation: \[(x^2-ax-bx+ab)(a+b)=(x-a)(x-b)(a+b)\] Simplify the right side of the equation: \(x^2a+ax^2-(2abx)+(a^2b^2)=(x-a)(x-b)(a+b)\) Now, equate the left side and the right side: \(2x^2ab-2a^2bx-2ab^2x+2a^2b^2=x^2a+ax^2-(2abx)+(a^2b^2)\)

Solve for x.

Since the equation is for x: \(x=\frac{-a^2b^2}{ab}+ab\) \(x=a+b\) The second part of the answer is when the fractions on the left side equal 0, which is when x = 0. So, the solution set is \(\{a+b, 0\}\), which corresponds to option (c).

Key concepts

Algebraic Fractions

Algebraic fractions are just like regular fractions, but instead of integers, they have algebraic expressions in the numerator and the denominator. Consider them as fractions dressed in letters and numbers. Less intimidating when you understand them, algebraic fractions follow the same rules as the fractions you already know.

When solving equations with algebraic fractions, finding a common denominator is often necessary to combine them, just like you would when adding or subtracting numerical fractions. This step creates a single fraction, making it easier to manage the next moves— simplification or cross multiplication—bringing you closer to the solution.

Common Denominator

Finding a common denominator is like organizing a party and making sure everyone has a common interest to talk about. In mathematics, it's crucial when you've got multiple fractions. It allows you to add, subtract, or compare fractions by giving them a shared base to stand on.

Here's how it works: you look for a number or expression that both denominators can divide into, often it's the product of the denominators if they have no common factors. It's a critical unifying step in solving equations involving fractions because once you've got your fractions standing on the same ground, you can combine them effortlessly.

Cross Multiplication

Cross multiplication is like a bridge that helps you get from one side of the equation to the other without getting wet. It's tremendously useful when you have an equation with two fractions separated by an equal sign.

In practice, you multiply the numerator of one fraction by the denominator of the other, doing the same in reverse, effectively 'crossing' them. The goal is to eliminate the fractions by moving away from division, allowing you to manipulate the equation more freely without the constraints of denominators. Remember, the rule is: multiply across, not up and down.

Equation Simplification

Simplifying an equation is like tidying up your room so you can find your stuff more easily. It's about taking an equation with complex expressions and making it neat and manageable.

The process often involves combining like terms, reducing fractions, or factoring expressions to find the essential parts that allow you to solve for the unknown variable. Simplification can involve a number of steps but the goal is clear—make the equation easier to understand and solve.