Problem 35 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Give an example of a rational equation that you would solve using cross multiplication and one that you would solve using the LCD. Explain your reasoning.

Step-by-Step Solution

Answer

A rational equation where cross multiplication would be used: \(\frac{x}{5} = \frac{3}{2}\). Cross multiplication is efficient here due to the equal ratio between both sides. An example where the LCD method would be used is: \(\frac{x}{2} + \frac{3}{4} = \frac{5}{8}\). Here, the denominators are different, so the LCD method is appropriate to use.

Step 1: Choose an Example for Cross Multiplication

Firstly, for cross multiplication, choose a rational equation of the form: \(\frac{x}{5} = \frac{3}{2}\). In this form, the equal ratio between the numerator and the denominator on both sides fits the direct application of cross multiplication.

Step 2: Explain Cross Multiplication

Cross multiplication involves multiplying the numerator of one ratio with the denominator of the other ratio, and vice-versa. So in this case, you would cross multiply to give \(2x = 15\). Solving for \(x\) gives \(x = \frac{15}{2}\). Hence, this method is efficient for this type of equation.

Step 3: Choose an Example for LCD method

Thirdly, for the LCD method, choose a rational equation of the form: \(\frac{x}{2} + \frac{3}{4} = \frac{5}{8}\). For this equation, the denominators are different and you can't just cross multiply here. It's necessary to create common ground by finding the LCD.

Step 4: Explain LCD method

You have to first find the LCD, which is 8 in this case, then you multiply each term by the LCD. This yields \(4x + 6 = 5\). Solving for \(x\) gives \(x = -\frac{1}{4}\). Hence, the LCD method provides a more efficient means of solving in this scenario.

Key Concepts

Cross Multiplication

Cross multiplication is a convenient method used in solving rational equations where two fractions are set equal to each other. Imagine you're faced with an equation like \(\frac{x}{5} = \frac{3}{2}\). This setup is perfect for cross multiplication because each side of the equation has a single fraction.

To perform cross multiplication, you multiply the numerator of one fraction by the denominator of the other, and do the same in reverse, effectively 'crossing' them. Here, you would multiply \(x\) by \(2\) and \(5\) by \(3\). This provides the new equation \(2x = 15\). After the cross multiplication is done, you simply solve the resulting linear equation. In this case, dividing both sides by \(2\) leads you to the solution: \(x = \frac{15}{2}\).

Why Use Cross Multiplication?

The main advantage of cross multiplication lies in its simplicity. It eliminates the fractions right away and reduces the problem to a basic algebraic equation, which is often easier to solve. It's particularly effective when each side of the equation contains exactly one fraction.

Least Common Denominator (LCD)

When solving equations with multiple fractions that have different denominators, the least common denominator (LCD) is an essential concept. It's the smallest number that each of the denominators can divide into. Knowing how to find and use the LCD is key to streamlining the process of solving complex rational equations.

Consider an equation like \(\frac{x}{2} + \frac{3}{4} = \frac{5}{8}\). With differing denominators, cross multiplication won’t work here. Instead, identify the smallest number that 2, 4, and 8 all evenly divide into, which is 8, to our advantage.

Each term is then multiplied by 8, transforming the equation into \(4x + 6 = 5\). Multiplying by the LCD transforms the equation into one without fractions, making it more manageable. After simplification, the algebraic equation can be solved normally, leading in this case to \(x = -\frac{1}{4}\).

Importance of the LCD

Using the LCD makes it possible to combine fractions by creating a common ground for them. Not only does this aid in simplifying and solving equations, but it's also a fundamental skill for working with complex rational expressions.

Algebraic Methods

A variety of algebraic methods exist for solving equations, each tailored to handle specific types of problems. In addition to cross multiplication and finding the LCD, methods such as factoring, completing the square, and using the quadratic formula are prominent depending on the equation at hand.

Algebraic methods generally start with simplifying expressions, like combining like terms and reducing fractions. This simplification process is followed by manipulating the equation to isolate the variable of interest.

Choosing the Right Method

When approaching an algebra problem, choosing the most efficient method is crucial. For simple linear equations, basic operations are sufficient. Cross multiplication and finding the LCD are effective for rational equations. For quadratic equations or polynomials, factoring or using the quadratic formula might be more appropriate. Selecting the right tool from the algebraic toolbox is essential to efficiently find the solution.

Equation Solving

Equation solving is a fundamental aspect of algebra that involves finding the values of variables that satisfy a given mathematical statement. The first step is typically to simplify the equation by eliminating unnecessary complexity. This can be done through methods such as combining like terms, distributing, or canceling.

Once simplified, the next aim is to get the variable on one side of the equation, with the goal of isolating it. For equations involving fractions or rational expressions, cross multiplication and the LCD method are vital tools.

Systematic Approach to Solving Equations

To ensure a consistent solving process, especially with more complex equations, it's important to approach them systematically. Whether you choose substitution, elimination, or graphical methods for systems of equations, or involve derivatives for calculus problems, maintaining clarity and a methodical approach is key. Always check your solution by substituting it back into the original equation to verify its correctness.