Problem 16 - Algebra 1: Concepts and Skills

Question

Find the missing numerator. $$ \frac{11}{3 x}=\frac{2}{12 x^{3}} $$

Step-by-Step Solution

Answer

The missing numerator is \(x^{2}=\frac{6}{132}\)

Step 1: Set up the equation

Given that \(\frac{11}{3x}=\frac{2}{12x^{3}}\), we can start by setting the two fractions equal to each other to find the missing numerator.

Step 2: Cross multiply

Cross multiply to get rid of the denominators, we multiply the numerator of the first fraction with the denominator of the second fraction and vice versa. This results in \(11*(12x^{3}) = 2*(3x)\).

Step 3: Simplify the equation

Simplify the equation to \(132x^{3} = 6x\).

Step 4: Solve for x

To solve for x, divide both sides of the equation by x. This results in \(132x^{2} = 6\). Then, divide both sides by 132. This gives \(x^{2}=\frac{6}{132}\)

Key Concepts

Cross Multiplication

Cross multiplication is a technique used in solving equations involving algebraic fractions. It's especially helpful when you have two fractions set equal to one another. By cross multiplying, you eliminate the fractions, making it easier to solve for the unknown variable.
Here's how it works:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Then, take the numerator of the second fraction and multiply it by the denominator of the first fraction.

This process gets rid of the fractions by creating a single equation with no denominators. For example, in the exercise provided, after cross multiplying \( rac{11}{3x} = rac{2}{12x^{3}} \), we end up with \( 11 \times 12x^{3} = 2 \times 3x \). This simplifies the problem and puts us on the right path to finding the solution.

Equation Solving

Solving equations, particularly when they involve variables, is foundational in algebra. Once you've set up the equation through steps like cross multiplication, your next aim is to isolate the variable.
To solve such equations, follow these general steps:

• Simplify both sides: eliminate any like terms if necessary and simplify what's left after cross multiplying.
• Isolate the variable: rearrange the equation to get the variable by itself. This often involves dividing or factoring out terms.
• Solve for the variable: once isolated, solve the equation for the variable, as seen in step 4 of the exercise.

In the given exercise, we simplified the equation \( 132x^{3} = 6x \) to \( 132x^{2} = 6 \) by dividing both sides by \( x \). Then, further simplified by dividing each side by 132, resulting in \( x^{2} = \frac{6}{132} \). Each step simplifies the problem, making it easier to pinpoint the variable and find the correct answer.

Rational Equations

Rational equations involve fractions with variables in their numerators or denominators—or both. These types of equations are common in algebra and often require techniques like cross multiplication for solutions.

When working with rational equations, it's key to:

• Identify the rational expressions: pinpoint where the variable is in the fractions.
• Clear the denominators: using cross multiplication is especially handy here because it eliminates fractions from the equation.
• Simplify the resulting polynomial equation: reduce it to its simplest form to solve for the variable.

For the rational equation provided \( \frac{11}{3x} = \frac{2}{12x^{3}} \), understanding that these denominators contain the variable \( x \) is crucial. Clearing them by cross multiplication gives us a simpler polynomial form \( 132x^{3} = 6x \), which we can then solve using regular algebraic methods. Rational equations might seem complex, but by systematically breaking them down into simpler components, they become much more manageable.