Solve The Equation:${ 2(x + 3) = 3x + 5 }$

Mar 11, 2025 by ADMIN 44 views

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

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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 linear equation, $2(x + 3) = 3x + 5$ . We will break down the solution into manageable steps, making it easy for readers to understand and follow along.

Understanding the Equation

The given equation is $2(x + 3) = 3x + 5$ . To solve this equation, we need to isolate the variable $x$ on one side of the equation. The equation is a linear equation, meaning it is in the form of $ax + b = cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Distributive Property

The first step in solving the equation is to apply the distributive property. The distributive property states that for any numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . We can apply this property to the left side of the equation:

2(x + 3) = 2x + 6

Now the equation becomes:

2x + 6 = 3x + 5

Subtracting 2x from Both Sides

The next step is to get all the terms with $x$ on one side of the equation. We can do this by subtracting $2x$ from both sides of the equation:

6 = x + 5

Subtracting 5 from Both Sides

Now we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 5 from both sides of the equation:

1 = x

Conclusion

In this article, we solved the linear equation $2(x + 3) = 3x + 5$ using the distributive property and basic algebraic manipulations. We broke down the solution into manageable steps, making it easy for readers to understand and follow along. By following these steps, readers can solve similar linear equations with ease.

Tips and Tricks

When solving linear equations, always start by applying the distributive property to the left side of the equation.
Use basic algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, to isolate the variable on one side of the equation.
Always check your solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics and Engineering: Linear equations are used to model the motion of objects, forces, and energies.
Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Not applying the distributive property: Failing to apply the distributive property can lead to incorrect solutions.
Not isolating the variable: Failing to isolate the variable on one side of the equation can lead to incorrect solutions.
Not checking the solution: Failing to check the solution by plugging it back into the original equation can lead to incorrect solutions.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, readers can solve linear equations with ease. Remember to always apply the distributive property, isolate the variable, and check the solution to ensure accuracy.

Final Thoughts

Solving linear equations is not just a mathematical concept; it has numerous real-world applications. By mastering the skills outlined in this article, readers can apply linear equations to solve problems in physics, engineering, economics, and computer science.

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It is a simple equation that can be solved using basic algebraic manipulations.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . It is used to expand expressions with parentheses.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

Apply the distributive property to the left side of the equation.
Isolate the variable on one side of the equation by adding or subtracting the same value to both sides.
Check the solution by plugging it back into the original equation.

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

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

Not applying the distributive property
Not isolating the variable
Not checking the solution

Q: How do I check my solution?

A: To check your solution, plug it back into the original equation and simplify. If the equation is true, then your solution is correct.

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 the motion of objects, forces, and energies.
Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Q: Can I use linear equations to solve problems in other areas?

A: Yes, linear equations can be used to solve problems in other areas, including:

Business: Linear equations can be used to model business problems, such as profit and loss.
Science: Linear equations can be used to model scientific problems, such as population growth.
Social Sciences: Linear equations can be used to model social science problems, such as economic growth.

Q: How do I know if an equation is linear or not?

A: An equation is linear if the highest power of the variable(s) is 1. If the highest power is greater than 1, then the equation is not linear.

Q: Can I use linear equations to solve systems of equations?

A: Yes, linear equations can be used to solve systems of equations. A system of equations is a set of two or more equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, follow these steps:

Write the system of equations in the form of $ax + by = c$ .
Use the method of substitution or elimination to solve for one variable.
Substitute the value of the variable into one of the original equations to solve for the other variable.

Q: What are some common methods for solving systems of linear equations?

A: Some common methods for solving systems of linear equations include:

Substitution Method: This method involves substituting the value of one variable into one of the original equations to solve for the other variable.
Elimination Method: This method involves adding or subtracting the same value to both sides of the equations to eliminate one variable.

Q: Can I use linear equations to solve quadratic equations?

A: Yes, linear equations can be used to solve quadratic equations. A quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: How do I solve a quadratic equation using linear equations?

A: To solve a quadratic equation using linear equations, follow these steps:

Write the quadratic equation in the form of $ax^2 + bx + c = 0$ .
Use the method of substitution or elimination to solve for one variable.
Substitute the value of the variable into one of the original equations to solve for the other variable.

Q: What are some common methods for solving quadratic equations?

A: Some common methods for solving quadratic equations include:

Factoring Method: This method involves factoring the quadratic expression into the product of two binomials.
Quadratic Formula Method: This method involves using the quadratic formula to solve for the variable.

Q: Can I use linear equations to solve polynomial equations?

A: Yes, linear equations can be used to solve polynomial equations. A polynomial equation is an equation in which the highest power of the variable(s) is greater than 2.

Q: How do I solve a polynomial equation using linear equations?

A: To solve a polynomial equation using linear equations, follow these steps:

Write the polynomial equation in the form of $ax^n + bx^{n-1} + \ldots + c = 0$ .
Use the method of substitution or elimination to solve for one variable.
Substitute the value of the variable into one of the original equations to solve for the other variable.

Q: What are some common methods for solving polynomial equations?

A: Some common methods for solving polynomial equations include:

Factoring Method: This method involves factoring the polynomial expression into the product of two or more binomials.
Synthetic Division Method: This method involves using synthetic division to divide the polynomial expression by a linear factor.

Conclusion

In conclusion, linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. By following the steps outlined in this article, readers can solve linear equations with ease. Remember to always apply the distributive property, isolate the variable, and check the solution to ensure accuracy.