Problem 1 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x}{6}=\frac{3}{2}$$

Step-by-Step Solution

Answer

The solution is \( x = 9 \).

Step 1: Understand the Equation

The equation given is \( \frac{x}{6}=\frac{3}{2} \). This is a simple equation involving fractions, where we need to solve for \( x \).

Step 2: Cross-Multiply to Eliminate Fractions

To solve the equation involving fractions, we can cross-multiply. Multiply both sides by 6 to eliminate the fraction on the left side: \[ x = \frac{3}{2} \times 6 \]

Step 3: Calculate the Right Side

Perform the multiplication on the right side: \[ x = 6 \times \frac{3}{2} = \frac{18}{2} \]

Step 4: Simplify the Fraction

Now, simplify the fraction by dividing 18 by 2:\[ x = 9 \]

Step 5: Verify the Solution

To ensure the solution is correct, substitute \( x = 9 \) back into the original equation:\[ \frac{9}{6} = \frac{3}{2} \]Dividing both the numerator and the denominator by 3, \( \frac{9}{6} = \frac{3}{2} \), which confirms that \( x = 9 \) is the correct solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is a handy technique used to simplify equations involving fractions. When you encounter an equation like \( \frac{x}{6} = \frac{3}{2} \), you need to remove the fractions to more easily solve for the unknown variable. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. This eliminates the fractions, making the equation easier to work with. In our equation, you would multiply \( x \) by 2 and 3 by 6, which gives:

• \( 2x = 18 \)

Now, solve for \( x \) by dividing both sides by 2, to isolate \( x \) on one side of the equation.

Fractions

Fractions represent parts of a whole and they are formed by a numerator and a denominator. In the equation \( \frac{x}{6} = \frac{3}{2} \), both sides of the equation are fractions. The numerator of a fraction is the number above the fraction bar and it tells how many parts we have. The denominator is the number below the fraction bar, and it shows the total number of equal parts the whole is divided into. Fractions can seem tricky at first, but by understanding how to manipulate them, including simplifying or cross-multiplying, they become much easier to deal with in equations.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. This often involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, after cross-multiplying in our equation, you end up with \( \frac{18}{2} \). To simplify \( \frac{18}{2} \), you divide both 18 and 2 by 2, their GCD:

• 18 divided by 2 is 9
• 2 divided by 2 is 1

Therefore, \( \frac{18}{2} = 9 \) simplifies to 9. Simplifying makes calculations easier and helps verify the accuracy of solutions, as seen when checking \( \frac{9}{6} \) by reducing it to \( \frac{3}{2} \). Knowing how to simplify is essential for solving equations efficiently.