Solving The Equation (1/2)(x+10)=(3/4)(x-2) A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. This article provides a comprehensive guide on how to solve the equation ${\frac{1}{2}(x+10)=\frac{3}{4}(x-2)}$. We'll walk through each step in detail, ensuring a clear understanding of the process. From basic principles to advanced techniques, you'll gain the knowledge and confidence to tackle a wide range of equations. Mastering equation solving is crucial for various applications in science, engineering, economics, and everyday problem-solving.

Understanding the Basics of Equations

Before diving into the specific equation, let's establish a solid foundation by understanding the basics of equations. An equation is a mathematical statement that asserts the equality of two expressions. It typically involves variables (represented by letters) and constants (numbers), connected by mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. These values are called solutions or roots of the equation.

To solve an equation effectively, it is important to understand the properties of equality. These properties allow us to manipulate equations while preserving their balance. The most commonly used properties include:

• Addition Property of Equality: If ${a = b}$, then ${a + c = b + c}$ for any number ${c}$.
• Subtraction Property of Equality: If ${a = b}$, then ${a - c = b - c}$ for any number ${c}$.
• Multiplication Property of Equality: If ${a = b}$, then ${ac = bc}$ for any number ${c}$.
• Division Property of Equality: If ${a = b}$, then ${\frac{a}{c} = \frac{b}{c}}$ for any non-zero number ${c}$.
• Distributive Property: ${a(b + c) = ab + ac}$

These properties form the cornerstone of equation solving, enabling us to isolate variables and find solutions. Let's move on to the first step in tackling our equation.

Step 1: Eliminate Fractions

The given equation is ${\frac{1}{2}(x+10)=\frac{3}{4}(x-2)}$. The presence of fractions can make the equation look intimidating, but we can eliminate them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 2 and 4. The LCM of 2 and 4 is 4. Therefore, we multiply both sides of the equation by 4: ${4 \times \frac{1}{2}(x+10) = 4 \times \frac{3}{4}(x-2)}$

Multiplying the left side by 4: ${4 \times \frac{1}{2}(x+10) = 2(x+10)}$

Multiplying the right side by 4: ${4 \times \frac{3}{4}(x-2) = 3(x-2)}$

Now our equation simplifies to: ${2(x+10) = 3(x-2)}$

By eliminating fractions, we have transformed the equation into a simpler form that is easier to work with. This is a crucial first step in solving many equations involving fractions. Next, we'll apply the distributive property to further simplify the equation.

Step 2: Apply the Distributive Property

In this step, we will apply the distributive property to remove the parentheses in the equation ${2(x+10) = 3(x-2)}$. The distributive property states that ${a(b + c) = ab + ac}$. Applying this property to the left side of the equation: ${2(x+10) = 2 \times x + 2 \times 10 = 2x + 20}$

Applying the distributive property to the right side of the equation: ${3(x-2) = 3 \times x - 3 \times 2 = 3x - 6}$

Now our equation becomes: ${2x + 20 = 3x - 6}$

The distributive property has helped us to expand the expressions and eliminate the parentheses. This simplification brings us closer to isolating the variable ${x}$. In the next step, we will focus on grouping like terms together.

Step 3: Group Like Terms

Our equation currently reads ${2x + 20 = 3x - 6}$. To solve for ${x}$, we need to group the terms containing ${x}$ on one side of the equation and the constant terms on the other side. Let's start by subtracting ${2x}$ from both sides of the equation: ${2x + 20 - 2x = 3x - 6 - 2x}$

This simplifies to: ${20 = x - 6}$

Now, we want to isolate the variable ${x}$ completely. To do this, we add 6 to both sides of the equation: ${20 + 6 = x - 6 + 6}$

This simplifies to: ${26 = x}$

Grouping like terms is a fundamental technique in solving equations. By strategically adding or subtracting terms from both sides, we can rearrange the equation to isolate the variable we are solving for. In this case, we have isolated ${x}$ on one side, but it is usually written with the variable on the left side.

Step 4: Isolate the Variable and Solve

From the previous step, we have arrived at the equation ${26 = x}$. While this is technically the solution, it is conventional to write the variable on the left side. So, we can simply rewrite the equation as: ${x = 26}$

We have now successfully isolated the variable ${x}$ and found its value. The solution to the equation ${\frac{1}{2}(x+10)=\frac{3}{4}(x-2)}$ is ${x = 26}$. Isolating the variable is the ultimate goal in solving equations. It involves performing operations on both sides of the equation until the variable is by itself on one side, revealing its value.

Step 5: Verify the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the value of ${x}$ back into the original equation. Our original equation is ${\frac{1}{2}(x+10)=\frac{3}{4}(x-2)}$, and our solution is ${x = 26}$. Substituting ${x = 26}$ into the original equation: ${\frac{1}{2}(26+10) = \frac{3}{4}(26-2)}$

Simplify the expressions inside the parentheses: ${\frac{1}{2}(36) = \frac{3}{4}(24)}$

Perform the multiplication: ${18 = 18}$

Since both sides of the equation are equal, our solution ${x = 26}$ is correct. Verifying the solution is a crucial step to prevent errors and gain confidence in our answer. It ensures that the value we found for the variable indeed satisfies the original equation.

Conclusion: Mastering Equation Solving

In this comprehensive guide, we have walked through the process of solving the equation ${\frac{1}{2}(x+10)=\frac{3}{4}(x-2)}$. We started by understanding the basics of equations and the properties of equality. Then, we followed a step-by-step approach:

1. Eliminated fractions by multiplying both sides by the LCM of the denominators.
2. Applied the distributive property to remove parentheses.
3. Grouped like terms by adding or subtracting terms from both sides.
4. Isolated the variable to find its value.
5. Verified the solution by substituting it back into the original equation.

By mastering these steps, you can confidently solve a wide range of linear equations. Remember, practice is key to improving your equation-solving skills. The more you practice, the more comfortable and proficient you will become. Solving equations is a fundamental skill in mathematics and has numerous applications in various fields. By mastering this skill, you equip yourself with a powerful tool for problem-solving and critical thinking. Keep practicing, and you'll become a master of equation solving!

This step-by-step approach not only helps in solving the specific equation but also provides a framework for tackling other algebraic problems. Understanding these fundamental concepts and practicing regularly will solidify your mathematical foundation and enhance your problem-solving capabilities. Happy solving!