Solve The Equation:${ 3(x - 7) = 5(2x + 1) }$

Mar 12, 2025 by ADMIN 47 views

Solving Linear Equations: A Step-by-Step Guide to Simplifying and Isolating Variables

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, $3(x - 7) = 5(2x + 1)$ , and provide a step-by-step guide on how to simplify and isolate the variable.

What are Linear Equations?

Linear equations are algebraic equations in which the highest power of the variable(s) is 1. They 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 $3(x - 7) = 5(2x + 1)$ . This equation involves two variables, $x$ and the constants $3$ , $5$ , $7$ , and $1$ . Our goal is to isolate the variable $x$ and find its value.

Step 1: Distribute the Numbers Outside the Parentheses

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

3(x - 7) = 3x - 21
5(2x + 1) = 10x + 5

Step 2: Rewrite the Equation with the Distributed Terms

Now that we have distributed the numbers outside the parentheses, we can rewrite the equation with the new terms.

3x - 21 = 10x + 5

Step 3: Add or Subtract the Same Value to Both Sides

To isolate the variable $x$ , we need to get all the terms with $x$ on one side of the equation. We can do this by adding or subtracting the same value to both sides of the equation.

3x - 21 + 21 = 10x + 5 + 21

Step 4: Simplify the Equation

Now that we have added the same value to both sides of the equation, we can simplify the equation by combining like terms.

3x = 10x + 26

Step 5: Subtract the Same Value from Both Sides

To isolate the variable $x$ , we need to get all the terms with $x$ on one side of the equation. We can do this by subtracting the same value from both sides of the equation.

3x - 10x = 26

Step 6: Simplify the Equation

Now that we have subtracted the same value from both sides of the equation, we can simplify the equation by combining like terms.

-7x = 26

Step 7: Divide Both Sides by the Coefficient

To find the value of $x$ , we need to divide both sides of the equation by the coefficient of $x$ .

x = -26/7

Conclusion

Solving linear equations requires a step-by-step approach, and in this article, we have walked through the process of solving the equation $3(x - 7) = 5(2x + 1)$ . By distributing the numbers outside the parentheses, rewriting the equation, adding or subtracting the same value to both sides, simplifying the equation, subtracting the same value from both sides, and dividing both sides by the coefficient, we have isolated the variable $x$ and found its value.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
When distributing numbers outside the parentheses, make sure to multiply the number by each term inside the parentheses.
When adding or subtracting the same value to both sides of the equation, make sure to add or subtract the same value to both sides.
When simplifying the equation, make sure to combine like terms.
When dividing both sides of the equation by the coefficient, make sure to divide both sides by the coefficient.

Real-World Applications

Solving linear equations has numerous real-world applications, including:

Physics: Solving linear equations is essential in physics to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Solving linear equations is crucial in engineering to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
Economics: Solving linear equations is essential in economics to model and analyze economic systems, including supply and demand, production, and consumption.
Computer Science: Solving linear equations is crucial in computer science to develop algorithms and data structures, including sorting, searching, and graph algorithms.

Final Thoughts

Solving linear equations is a fundamental skill that has numerous real-world applications. By following the step-by-step guide outlined in this article, you can solve linear equations with ease and confidence. Remember to distribute the numbers outside the parentheses, rewrite the equation, add or subtract the same value to both sides, simplify the equation, subtract the same value from both sides, and divide both sides by the coefficient to isolate the variable. With practice and patience, you can become proficient in solving linear equations and apply your skills to real-world problems.

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Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. In this article, we will provide a Q&A guide to help you understand the process of solving linear equations.

Q: What is a linear equation?

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

Q: What are the steps involved in solving a linear equation?

A: The steps involved in solving a linear equation are:

Distribute the numbers outside the parentheses to the terms inside.
Rewrite the equation with the distributed terms.
Add or subtract the same value to both sides of the equation.
Simplify the equation by combining like terms.
Subtract the same value from both sides of the equation.
Divide both sides of the equation by the coefficient.

Q: What is the order of operations (PEMDAS) and how does it apply to solving linear equations?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

When solving linear equations, it's essential to follow the order of operations to ensure that you are performing the operations in the correct order.

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

A: A linear equation is an algebraic equation in which the highest power of the variable(s) is 1. A quadratic equation, on the other hand, is an algebraic equation in which the highest power of the variable(s) is 2. Quadratic equations can be written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

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

A: Solving linear equations has numerous real-world applications, including:

Physics: Solving linear equations is essential in physics to describe the motion of objects, including velocity, acceleration, and force.
Engineering: Solving linear equations is crucial in engineering to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
Economics: Solving linear equations is essential in economics to model and analyze economic systems, including supply and demand, production, and consumption.
Computer Science: Solving linear equations is crucial in computer science to develop algorithms and data structures, including sorting, searching, and graph algorithms.

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

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

Not following the order of operations (PEMDAS)
Not distributing the numbers outside the parentheses to the terms inside
Not rewriting the equation with the distributed terms
Not adding or subtracting the same value to both sides of the equation
Not simplifying the equation by combining like terms
Not subtracting the same value from both sides of the equation
Not dividing both sides of the equation by the coefficient

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. By following the Q&A guide provided in this article, you can gain a better understanding of the process of solving linear equations and avoid common mistakes. Remember to distribute the numbers outside the parentheses, rewrite the equation, add or subtract the same value to both sides, simplify the equation, subtract the same value from both sides, and divide both sides by the coefficient to isolate the variable. With practice and patience, you can become proficient in solving linear equations and apply your skills to real-world problems.