Solve The Equation For { X$} : : : { 5(x + 2) = 3(x + 8) \}

Mar 9, 2025 by ADMIN 60 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which is a simple equation involving one variable. We will use the equation 5(x + 2) = 3(x + 8) as an example to demonstrate the step-by-step process of solving linear equations.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to be Solved

The equation we will be solving is 5(x + 2) = 3(x + 8). This equation involves two variables, x and the constants 5, 2, 3, and 8. Our goal is to isolate the variable x and find its value.

Step 1: Distribute the Numbers

To solve the equation, we need to start by distributing the numbers outside the parentheses to the terms inside. This means multiplying the number outside the parentheses by each term inside.

5(x + 2) = 5x + 10
3(x + 8) = 3x + 24

Step 2: Simplify the Equation

Now that we have distributed the numbers, we can simplify the equation by combining like terms. In this case, we have two terms on each side of the equation: 5x and 10 on the left side, and 3x and 24 on the right side.

5x + 10 = 3x + 24

Step 3: Isolate the Variable

Our goal is to isolate the variable x, which means getting x by itself on one side of the equation. To do this, we need to get rid of the constant term on the same side as x. We can do this by subtracting 3x from both sides of the equation.

5x - 3x + 10 = 3x - 3x + 24
2x + 10 = 24

Step 4: Solve for x

Now that we have isolated the variable x, we can solve for its value. To do this, we need to get rid of the constant term on the same side as x. We can do this by subtracting 10 from both sides of the equation.

2x + 10 - 10 = 24 - 10
2x = 14

Step 5: Divide to Find x

Finally, we can find the value of x by dividing both sides of the equation by 2.

2x / 2 = 14 / 2
x = 7

Conclusion

Solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article, we can solve equations involving one variable. Remember to distribute the numbers, simplify the equation, isolate the variable, and solve for its value. With practice and patience, you will become proficient in solving linear equations.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. These include:

Not distributing the numbers: Failing to distribute the numbers outside the parentheses to the terms inside can lead to incorrect solutions.
Not simplifying the equation: Failing to combine like terms can make the equation more complicated and difficult to solve.
Not isolating the variable: Failing to get the variable by itself on one side of the equation can make it difficult to solve for its value.
Not checking the solution: Failing to check the solution by plugging it back into the original equation can lead to incorrect solutions.

Tips and Tricks

When solving linear equations, there are several tips and tricks to keep in mind. These include:

Use algebraic manipulation: Algebraic manipulation is a powerful tool for solving linear equations. By using algebraic properties such as the distributive property, we can simplify the equation and isolate the variable.
Use graphing: Graphing is a visual way to solve linear equations. By graphing the equation, we can see the solution and verify that it is correct.
Use substitution: Substitution is a method of solving linear equations by substituting one variable for another. This can be a useful technique when solving equations involving multiple variables.
Check the solution: Always check the solution by plugging it back into the original equation. This ensures that the solution is correct and accurate.

Real-World Applications

Linear equations have numerous real-world applications. These include:

Physics and engineering: Linear equations are used to model real-world phenomena such as motion, force, and energy.
Economics: Linear equations are used to model economic systems and make predictions about future trends.
Computer science: Linear equations are used to solve problems in computer science such as linear programming and optimization.
Data analysis: Linear equations are used to analyze and interpret data in various fields such as medicine, social sciences, and business.

Conclusion

Introduction

Solving linear equations is a crucial skill for students to master. In our previous article, we provided a step-by-step guide on how to solve linear equations. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

Distribute the numbers outside the parentheses to the terms inside.
Simplify the equation by combining like terms.
Isolate the variable by getting it by itself on one side of the equation.
Solve for the value of the variable.

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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation 2x + 3 = 5 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your solution by plugging it back into the original equation to ensure that it's correct.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not distributing the numbers outside the parentheses to the terms inside.
Not simplifying the equation by combining like terms.
Not isolating the variable by getting it by itself on one side of the equation.
Not checking the solution by plugging it back into the original equation.

Q: Can I use algebraic manipulation to solve linear equations?

A: Yes, you can use algebraic manipulation to solve linear equations. Algebraic manipulation involves using algebraic properties such as the distributive property to simplify the equation and isolate the variable.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

Physics and engineering: Linear equations are used to model real-world phenomena such as motion, force, and energy.
Economics: Linear equations are used to model economic systems and make predictions about future trends.
Computer science: Linear equations are used to solve problems in computer science such as linear programming and optimization.
Data analysis: Linear equations are used to analyze and interpret data in various fields such as medicine, social sciences, and business.

Q: Can I use graphing to solve linear equations?

A: Yes, you can use graphing to solve linear equations. Graphing involves plotting the equation on a coordinate plane and finding the point of intersection to determine the solution.

Q: What are some tips and tricks for solving linear equations?

A: Some tips and tricks for solving linear equations include:

Use algebraic manipulation to simplify the equation and isolate the variable.
Use graphing to visualize the equation and find the solution.
Check the solution by plugging it back into the original equation to ensure that it's correct.
Use substitution to solve equations involving multiple variables.