Problem 40 - Prealgebra

Question

Solve each equation. $$\frac{x}{4}=\frac{6}{3}$$

Step-by-Step Solution

Answer

The solution is \( x = 8 \).

Step 1: Understand the equation format

The equation is given as \( \frac{x}{4} = \frac{6}{3} \). This is a proportion, which means two ratios are equal.

Step 2: Simplify the right-hand side fraction

Simplify the fraction on the right-hand side: \( \frac{6}{3} \) simplifies to 2 because 6 divided by 3 equals 2.

Step 3: Rewrite the equation after simplification

After simplification, the equation becomes \( \frac{x}{4} = 2 \).

Step 4: Solve for x

To solve for \( x \), multiply both sides of the equation by 4 to isolate \( x \) on one side:\( x = 2 \times 4 \).

Step 5: Compute the multiplication

Perform the multiplication: \( 2 \times 4 = 8 \). So, \( x = 8 \).

Key Concepts

Understanding Proportions

Proportions are equations that state that two ratios are equal. In this case, our equation is \( \frac{x}{4} = \frac{6}{3} \). Here, \( \frac{x}{4} \) and \( \frac{6}{3} \) are two ratios. Understanding proportions is fundamental as it allows us to compare different fractions and solve for unknown variables. A key point in solving proportions is ensuring that you work to maintain the equality between these two ratios.

When dealing with proportions, the primary objective is to maintain the balance between the left-side ratio and the right-side ratio. In simpler terms, you want to find a number that makes both sides equal.

Here's a quick tip: you can always check your proportion by cross-multiplying. For example, in \( \frac{x}{4} = \frac{6}{3} \), multiplying across the equals gives \( 3x = 24 \). Solving this will again confirm your solutions.

Simplifying Fractions

Fractions make up a significant part of mathematics, and simplifying them is an essential skill. In our problem, we were given \( \frac{6}{3} \) which we simplified to a whole number: 2.

Simplifying a fraction means reducing it to its simplest form. This occurs when the numerator and the denominator are coprime, sharing no common factors other than 1.

There are a few simple steps you can follow to simplify:

• Identify a common factor between the numerator and the denominator.
• Divide both numbers by the greatest common factor.

For \( \frac{6}{3} \), the greatest common factor is 3. Dividing both 6 and 3 by this number results in 2.

Remember, simplifying fractions makes them easier to understand and reduces computational complexity, which is particularly useful when trickier calculations crop up.

Isolation of Variables

In solving linear equations, one of the most crucial techniques is isolating the variable. This means getting the variable, in this case, \( x \), by itself on one side of the equation to solve it directly.

After simplifying the fraction, the equation becomes \( \frac{x}{4} = 2 \). To isolate \( x \), you multiply both sides by 4. This step removes 4 from the denominator, leaving \( x = 8 \).

The steps to isolate a variable are straightforward:

• Look for any coefficients or dividing terms attached to the variable.
• Move these to the other side of the equation using operations like multiplication or division.

This action essentially "annuls" these terms on the side with the variable, focusing the equation down to the variable’s value.

Mastering this process transforms complicated looking equations into simple, solvable expressions, making your math life so much easier!