Problem 5 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x}{3}=\frac{5}{2}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{15}{2} \).

Step 1: Understand the Equation

We are given the equation \( \frac{x}{3} = \frac{5}{2} \). This is a simple proportion where \( x \) is divided by 3, and it equals \( \frac{5}{2} \). Our goal is to solve for \( x \).

Step 2: Cross-Multiply to Eliminate Fractions

To solve for \( x \), we can eliminate the fractions by cross-multiplying: \( 2 \cdot x = 5 \cdot 3 \). This simplifies the equation, allowing us to solve directly for \( x \).

Step 3: Simplify the Result

After performing the cross-multiplication, we simplify the equation: \( 2x = 15 \).

Step 4: Solve for x

To isolate \( x \), divide both sides of the equation by 2: \( x = \frac{15}{2} \). This is our final solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. Equations are statements that express equality between two expressions. They can include numbers, variables (like \( x \)), and mathematical operations such as addition or multiplication. Our objective when solving an equation is to find the value of the variable that makes the equation true. This process often involves several steps:

• Simplify both sides: Break down complex expressions into more manageable parts by simplifying or combining like terms.
• Isolate the variable: Use inverse operations to get the variable by itself on one side of the equation. This usually involves adding, subtracting, multiplying, or dividing both sides of the equation.
• Check your solution: Substitute the value back into the original equation to ensure it satisfies the equation.

Using these steps helps to systematically solve for the unknown variable. Solving equations is essential in mathematics as it allows us to find specific values that satisfy mathematical models or real-world scenarios.
Understanding how to navigate and simplify each part of an equation is the key to becoming proficient at solving them.

Cross-Multiplication

Cross-multiplication is a technique used to eliminate fractions from an equation, making it easier to solve. When you have an equation in the form of a proportion, like \( \frac{x}{a} = \frac{b}{c} \), cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other and vice versa. Here's how it works in simple steps:

• Multiply the numerator of the first fraction (\( x \)) by the denominator of the second fraction (\( c \))
• Multiply the denominator of the first fraction (\( a \)) by the numerator of the second fraction (\( b \))
• Set these two products equal to each other: \( xc = ab \)

This method effectively removes the fractions, leaving you with a straightforward equation to solve. Cross-multiplication is especially useful in problems involving proportions where it simplifies the process of finding the unknown variable. It's a handy tool that saves time and simplifies calculations.

Proportions

Proportions describe a relationship between two ratios or fractions. A proportion states that two ratios are equivalent. For example, \( \frac{x}{y} = \frac{a}{b} \) expresses that the ratio of \( x \) to \( y \) is the same as the ratio of \( a \) to \( b \). Proportions are powerful in solving problems where one set of quantities is compared to another, allowing you to predict one value if others are known. Here are some common applications:

• Recipe Adjustments: If a recipe ratios ingredients to servings, you can adjust the numbers to serve more or fewer people while maintaining taste.
• Map Work: Map scales are proportions that help translate distances on a map to real-life distances.
• Scaling Models: Architects and engineers use proportions to create scale models, maintaining accurate relationships between dimensions.

Understanding proportions makes it easier to solve problems where two quantities scale relative to each other. It also provides a foundation for tackling more complex mathematical and real-world challenges.

Linear Equations

Linear equations are a type of equation where the highest power of the variable is 1. They can often be written in the form \( ax + b = c \), where \( x \) is the variable, \( a \), \( b \), and \( c \) are constants. These equations create straight lines when graphed, and are characterized by consistent rates of change. To solve a linear equation, you follow a series of steps:

• Combine like terms on each side of the equation if necessary.
• Move variable terms to one side of the equation and constants to the other.
• Divide or multiply to isolate the variable and solve for it.

Linear equations are fundamental in understanding algebraic concepts and are used in various fields such as physics, economics, and engineering to model linear relationships. Recognizing and working with linear equations is an essential skill since they often serve as the starting point for solving more complex algebraic problems.