Problem 28 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{x}{3}=72 $$

Step-by-Step Solution

Answer

Answer: The value of x in the equation $$\frac{x}{3} = 72$$ is 216.

Step 1: Identify the Equation

The equation to solve is given as: $$ \frac{x}{3} = 72 $$

Step 2: Multiply both sides by 3

To isolate the variable x, we need to get rid of the denominator of the fraction. Since x is divided by 3, we can multiply both sides of the equation by 3 to cancel the denominator: $$ 3 \cdot \frac{x}{3} = 72 \cdot 3 $$

Step 3: Simplify the Equation

After multiplying, the left side of the equation simplifies as follows, as the 3 cancels out in the fraction: $$ x = 72 \cdot 3 $$

Step 4: Calculate x

Now we just need to calculate the value of x by performing the multiplication on the right side of the equation: $$ x = 216 $$

Step 5: State the Solution

The value of x that makes the given conditional equation true is 216. So, the solution to the equation $$\frac{x}{3} = 72$$ is $$x=216$$.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, we use symbols to represent quantities without specifying their actual values.

• The basic idea is to use letters like \( x \), \( y \), and \( z \) to represent numbers.
• These letters are called variables because they can take on different values.

Algebra allows us to write equations that can be solved to find unknown values. For instance, the equation \( \frac{x}{3} = 72 \) involves solving for the variable \( x \). By using algebra, we ascertain the value of the unknown that satisfies this equation.

Multiplication

Multiplication is one of the four fundamental arithmetic operations, along with addition, subtraction, and division. It is essentially a quicker way to do repeated addition.

• When you multiply a number, you add it to itself a certain number of times.
• For example, \( 3 \times 72 \) means adding 72 three times: \( 72 + 72 + 72 \).

In the context of solving equations, multiplication can be used to eliminate fractions by multiplying through by the common denominator. In our equation \( \frac{x}{3} = 72 \), we multiply both sides by 3 to simplify, leading to \( x = 72 \times 3 \).

Fractions

Fractions represent a part of a whole and consist of two numbers: the numerator and the denominator.

• The numerator is the top number, indicating how many parts we have.
• The denominator is the bottom number, indicating into how many parts the whole is divided.

In \( \frac{x}{3} = 72 \), the fraction \( \frac{x}{3} \) represents \( x \) divided by 3. To clear the fraction for solving equations, we often multiply by the denominator to "cancel" it out. Here, multiplying by 3 isolates \( x \).

Isolating Variables

Isolating the variable is a core technique in algebra when solving equations. This process means rearranging the equation to have the variable on one side.

• You perform operations that "undo" what is being done to the variable.
• These operations are applied equally to both sides of the equation to maintain balance.

With \( \frac{x}{3} = 72 \), isolating \( x \) requires us to eliminate the fraction by multiplying both sides by 3, giving us \( x = 216 \). The goal of isolation is to find the numerical value of the variable that satisfies the equation.