Problem 67 - Algebra for College Students

Question

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{4}{x}=\frac{3}{4}$$

Step-by-Step Solution

Answer

x = \frac{16}{3}

Step 1: Understand the Equation

The equation given is \(\frac{4}{x}=\frac{3}{4}\). This means the two fractions are equal.

Step 2: Cross Multiply

To solve for \(x\), cross-multiply the fractions: \(4 \times 4 = 3 \times x\).

Step 3: Simplify the Multiplication

Simplify the multiplication on both sides: \(16 = 3x\).

Step 4: Solve for x

Divide both sides by 3 to isolate \(x\): \(x = \frac{16}{3}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations involving fractions. When you have an equation where two fractions are equal, you can use cross-multiplication to eliminate the fractions. This is done by multiplying the numerator of each fraction by the denominator of the other fraction.

For example, in the equation \(\frac{4}{x} = \frac{3}{4}\), you cross-multiply as follows:
\(4 \times 4 = 3 \times x\)
This results in the equation: \(16 = 3x\).

By cross-multiplying, you change the problem from one involving fractions to a simpler equation without fractions.

Isolating Variables

Isolating variables is an essential step in solving equations. The goal is to get the unknown variable by itself on one side of the equation. This often involves dividing or multiplying both sides of the equation by the same number.

In our example, after cross-multiplying, we got the equation: \(16 = 3x\).

To isolate the variable \(x\), we need to divide both sides by 3:
\(x = \frac{16}{3}\)

By isolating the variable, we find the value of \(x\). It’s a crucial technique and is used in many algebra problems.

Fraction Manipulation

Understanding how to manipulate fractions is crucial for solving rational equations. This includes knowing how to cross-multiply, add, subtract, multiply, and divide fractions.

In our problem, the fractions were already set equal to each other, so we used cross-multiplication. However, other problems may require different manipulations:

• Adding/Subtracting: Find a common denominator before combining the fractions.
• Multiplying: Multiply the numerators together and the denominators together.
• Dividing: Invert (flip) the second fraction and then multiply.

Being skilled at fraction manipulation allows you to simplify complex fractions and solve equations more easily.

Equation Solving Techniques

To tackle any equation, it’s important to use the right techniques. Here are some common methods:

• Identifying the type of equation you’re dealing with.
• Using cross-multiplication for rational equations.
• Isolating variables to solve for unknowns.
• Checking your solution by substituting it back into the original equation.

By following a structured approach, you can methodically work through problems and find solutions. In our example, we used cross-multiplication, simplified the result, and isolated the variable. These techniques are foundational skills in algebra that you’ll use regularly.