Problem 36 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{x}{2.19}=5 $$

Step-by-Step Solution

Answer

Question: Solve the conditional equation and find the value of x: $$\frac{x}{2.19} = 5$$ Answer: x = 10.95

Step 1: Multiply both sides by 2.19

To eliminate the denominator (2.19) on the left side, multiply both sides of the equation by 2.19: $$ 2.19 \cdot \frac{x}{2.19} = 5 \cdot 2.19 $$

Step 2: Simplify the left side

Simplify the left side by canceling the denominator (2.19) and numerator (x): $$ x = 5 \cdot 2.19 $$

Step 3: Calculate the value of x

Multiply 5 by 2.19 to find the value of x: $$ x = 10.95 $$ The solution to the conditional equation is x = 10.95.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. It is essential in solving problems where certain values are unknown. In algebra, we rely on different operations such as addition, subtraction, multiplication, and division to simplify and solve equations.
These operations help us find the unknown values by maintaining the balance of the equation. This is done by performing the same mathematical operation on both sides of the equation.

• Focus on how equations are structured using variables and constants.
• Understand the importance of maintaining equality across both sides of the equation.
• Learn to manipulate these equations using arithmetic operations.

The goal is to isolate the variable on one side of the equation and determine its value. In the given example, the equation is structured with a fraction involving a variable. This requires manipulating the equation to solve for the unknown factor.

Steps to Solving Equations

Solving equations involves using a sequence of steps systematically to reach the unknown value of the variable. Follow these easy instructions to solve basic algebraic equations:

• Identify the equation and the unknown variable to solve for.
• Apply inverse operations to isolate the variable. This means undoing anything that affects the variable directly.
• Simplify the equation by performing operations like multiplication or division to both sides of the equation.
• Verify the solution by substituting the value back into the original equation.

In the original example, we had an equation where dividing by 2.19 was part of finding the value of \(x\). By multiplying both sides by 2.19, we isolated \(x\) and found it was equal to 10.95. This process showcases how inverse operations help simplify equations and find the solution.

The Role of Mathematics Education

Mathematics education plays a crucial role in equipping students with the skills needed to solve real-world problems. It helps students develop critical thinking and reasoning abilities by working through mathematical concepts and problems. In this context, learning involves:

• Gaining a solid understanding of foundational mathematics like algebra and arithmetic.
• Encouraging a problem-solving mindset to tackle new or complex equations.
• Providing examples and exercises to practice solving different types of mathematical problems.

Effective mathematics education encourages students to understand the logic behind solving equations, rather than just memorizing procedures. By knowing why we perform certain steps in algebra, students can apply those steps to a variety of problems with confidence and precision. In the exercise we're discussing, students learn how to manipulate equations and gracefully arrive at the solution while building a deeper understanding of mathematical principles.