Solve The Equation For { X $} : : : { \frac{2x}{5} = \frac{9}{15} \}

Introduction

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we will focus on solving the equation $\frac{2x}{5} = \frac{9}{15}$ for the variable $x$. We will break down the solution into step-by-step instructions and provide a clear explanation of each step.

Understanding the Equation

Before we start solving the equation, let's take a closer look at it. The equation is $\frac{2x}{5} = \frac{9}{15}$. This equation states that the ratio of $2x$ to $5$ is equal to the ratio of $9$ to $15$. Our goal is to find the value of $x$ that makes this equation true.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To solve the equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of $5$ and $15$ is $15$. So, we multiply both sides of the equation by $15$.

$$\frac{2x}{5} \cdot 15 = \frac{9}{15} \cdot 15$$

This simplifies to:

$$6x = 9$$

Step 2: Divide Both Sides by the Coefficient of x

Now that we have eliminated the fractions, we can focus on solving for $x$. We can do this by dividing both sides of the equation by the coefficient of $x$, which is $6$.

$$\frac{6x}{6} = \frac{9}{6}$$

This simplifies to:

$$x = \frac{3}{2}$$

Conclusion

In this article, we solved the equation $\frac{2x}{5} = \frac{9}{15}$ for the variable $x$. We broke down the solution into two steps: multiplying both sides of the equation by the LCM of the denominators and dividing both sides of the equation by the coefficient of $x$. By following these steps, we were able to find the value of $x$ that makes the equation true.

Tips and Tricks

• When solving equations, it's essential to follow the order of operations (PEMDAS).
• Make sure to multiply both sides of the equation by the LCM of the denominators to eliminate the fractions.
• Divide both sides of the equation by the coefficient of $x$ to solve for $x$.

Real-World Applications

Solving equations is a fundamental concept in mathematics that has numerous real-world applications. Here are a few examples:

• Physics: Solving equations is essential in physics to describe the motion of objects and predict their behavior.
• Engineering: Solving equations is crucial in engineering to design and optimize systems, such as bridges and buildings.
• Economics: Solving equations is used in economics to model the behavior of economic systems and make predictions about future trends.

Common Mistakes to Avoid

• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving equations.
• Not multiplying both sides by the LCM: Make sure to multiply both sides of the equation by the LCM of the denominators to eliminate the fractions.
• Not dividing both sides by the coefficient of x: Make sure to divide both sides of the equation by the coefficient of $x$ to solve for $x$.

Conclusion

$$$$$$$$

Introduction

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In our previous article, we solved the equation $\frac{2x}{5} = \frac{9}{15}$ for the variable $x$. In this article, we will provide a Q&A guide to help you understand the concept of solving equations and provide answers to common questions.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It is a way of expressing a relationship between two or more variables.

Q: What is a variable?

A: A variable is a symbol or letter that represents a value that can change. In an equation, the variable is the value that we are trying to find.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when solving an equation. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I solve an equation with fractions?

A: To solve an equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 5 and 15 is 15.

Q: How do I solve an equation with decimals?

A: To solve an equation with decimals, you can multiply both sides of the equation by a power of 10 to eliminate the decimals.

Q: What is the coefficient of a variable?

A: The coefficient of a variable is the number that is multiplied by the variable. For example, in the equation 2x, the coefficient of x is 2.

Q: How do I solve an equation with multiple variables?

A: To solve an equation with multiple variables, you need to isolate one variable at a time. You can do this by using the order of operations and the rules for solving equations.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Conclusion

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we provided a Q&A guide to help you understand the concept of solving equations and provide answers to common questions. We hope that this guide has been helpful in understanding the concept of solving equations.

Tips and Tricks

• Make sure to follow the order of operations when solving equations.
• Eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.
• Use the quadratic formula to solve quadratic equations.
• Isolate one variable at a time when solving equations with multiple variables.

Real-World Applications

Solving equations is a fundamental concept in mathematics that has numerous real-world applications. Here are a few examples:

• Physics: Solving equations is essential in physics to describe the motion of objects and predict their behavior.
• Engineering: Solving equations is crucial in engineering to design and optimize systems, such as bridges and buildings.
• Economics: Solving equations is used in economics to model the behavior of economic systems and make predictions about future trends.

Common Mistakes to Avoid

• Not following the order of operations: Make sure to follow the order of operations when solving equations.
• Not eliminating fractions: Make sure to eliminate fractions by multiplying both sides of the equation by the LCM of the denominators.
• Not using the quadratic formula: Make sure to use the quadratic formula to solve quadratic equations.