Problem 6 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{32}{x}=\frac{16}{3} $$

Step-by-Step Solution

Answer

Question: Solve the following rational equation and find the value of x: $$ \frac{32}{x}=\frac{16}{3} $$ Answer: The solution for the given rational equation is x = 6.

Step 1: Analyze the equation with fractions

We have the equation: $$ \frac{32}{x}=\frac{16}{3} $$ Notice that the denominator cannot be zero, so x ≠ 0.

Step 2: Cross-multiply

To remove the fractions, we'll cross-multiply the numerators with the denominators from the other side of the equation: $$ 32 * 3 = 16 * x $$

Step 3: Distribute and simplify

Now we'll distribute and simplify the equation: $$ 96 = 16x $$

Step 4: Solve for x

To isolate x, divide both sides of the equation by 16: $$ \frac{96}{16} = x $$ $$ x=6 $$ So the solution for the given rational equation is x=6.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a useful technique for solving rational equations, especially those involving algebraic fractions. Imagine you have an equation with two fractions set equal to each other, like \( \frac{32}{x} = \frac{16}{3} \). Cross-multiplication involves multiplying the numerator of each fraction by the denominator of the opposite fraction.
This helps to eliminate the fractions and simplify the equation.
Here's how it works:

• Take the numerator of the first fraction (32) and multiply it by the denominator of the second fraction (3). The equation becomes: \(32 \times 3\).
• Then, take the numerator of the second fraction (16) and multiply it by the denominator of the first fraction (x). This gives you: \(16 \times x\).
• Finally, set the products equal to each other, resulting in \(32 \times 3 = 16 \times x\).

This method simplifies the two-sided equation without fractions, making it easier to solve for the unknown variable.

Solving Equations

Once the equation is free from fractions through cross-multiplication, the next step is solving for the unknown variable. Solving equations involves a few steps designed to isolate the variable, in this case, \(x\). Here's how you can do it:

• First, simplify the products from the cross-multiplication. In the equation \(32 \times 3 = 16 \times x\), simplify \(32 \times 3\) to get 96.
• Now the equation reads \(96 = 16x\). The objective is to isolate \(x\) on one side.
• To do this, divide both sides by 16: \(\frac{96}{16} = x\).
• Calculating \(\frac{96}{16}\), you'll find that \(x = 6\).

These steps effectively solve the equation and reveal that \(x=6\). Remember, always check if the solution makes the original denominators zero to ensure it's valid.

Concept of Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator, or both, contain algebraic expressions. In rational equations, like the one here \(\frac{32}{x} = \frac{16}{3}\), understanding algebraic fractions is crucial. It's important to know how to manipulate these fractions while solving equations.

• Algebraic fractions can sometimes become undefined, particularly when their denominators equal zero. That's why it's essential to note that \(x eq 0\) in this exercise.
• The process of solving equations involving algebraic fractions typically includes cross-multiplication, which helps to remove the fractions and simplifies the equation.
• Be confident in basic fraction operations, simplifying expressions, and understand how to perform operations such as addition, subtraction, multiplication, and division with variables in denominators.

Mastering these concepts helps in solving complex rational equations involving algebra and prevents errors when solutions are invalid, such as having the denominator value as zero.