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Algebraic equations are mathematical statements that express the equality between two algebraic expressions. Equations are an essential part of algebra, allowing us to find unknown values. An algebraic equation has variables and constants, the latter representing known values. The equation demonstrates the concept of balance, where what we do to one side must be done to the other. For example, if we need to solve for a variable in the equation, it involves rearranging terms so that the variable is isolated on one side. In our problem, the equation set up is \(X = 4\left(\frac{X}{3} - 75\right)\). The primary goal is to solve for \(X\), which represents the total monthly rent.

Cost sharing is the idea of dividing a total cost among multiple individuals who share a service or product. In this exercise, three students planned to share the apartment's rent equally, which means dividing the total cost into equal parts. By adding a fourth person, the cost for each individual decreases, making it more affordable for everyone involved. This concept is not only crucial in practical situations like renting, but also in collaborative settings where expenses are pooled and split. Adding people to share costs can make large expenses more manageable, illustrating the benefits of cost sharing. Each student saves $75, demonstrating how adding the fourth person impacts cost per person.

Variables are symbols used in mathematics to represent unknown values or values that can change. They are fundamental in algebra, allowing us to formulate and solve equations systematically. In our problem, \(X\) represents the total monthly rent that needs to be shared. The variable simplifies the communication of mathematical problems, making it easier to manipulate and solve equations. Additionally, variables help convert real-world problems into mathematical expressions, enabling easier analysis and solution finding. Understanding the role of variables is crucial, because they allow us to express general relationships and solve problems that involve unknown quantities.

The process of solving equations involves several clear and logical steps. Initially, you identify the variable that needs to be determined, as shown by using \(X\) for the monthly rent. Once the equation \(X = 4\left(\frac{X}{3} - 75\right)\) is formulated, you start by applying mathematical operations to simplify and solve it.

• Distribute any constants, such as the 4 on the right-hand side to get \( 4\left(\frac{X}{3} - 75\right) = \frac{4}{3}X - 300 \).
• Rearrange the equation to isolate variable terms on one side, bringing like terms together: \(X = \frac{4}{3}X - 300\).
• Solve for the variable by performing operations that isolate it, such as subtracting \(\frac{4}{3}X\) from both sides to obtain \(\frac{1}{3}X = 300\).
• Finally, multiply by the reciprocal to find \(X = 900\).

These steps demonstrate how structured operations lead to solving for unknowns proficiently.