Solve $2x - 5 = 7$.

Mar 10, 2025 by ADMIN 22 views

Solve the Linear Equation $2x - 5 = 7$

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Introduction

In this article, we will focus on solving a linear equation, specifically the equation $2x - 5 = 7$. Linear equations are a fundamental concept in mathematics, and solving them is an essential skill for students and professionals alike. The equation $2x - 5 = 7$ is a simple linear equation that can be solved using basic algebraic techniques.

Understanding the Equation

The equation $2x - 5 = 7$ is a linear equation in one variable, which means it has only one unknown variable, x. The equation is in the form of ax + b = c, where a, b, and c are constants. In this case, a = 2, b = -5, and c = 7.

Step 1: Add 5 to Both Sides

To solve the equation, we need to isolate the variable x. The first step is to add 5 to both sides of the equation, which will eliminate the negative term. This can be done as follows:

2x - 5 + 5 = 7 + 5

2x = 12

Step 2: Divide Both Sides by 2

Now that we have isolated the term with the variable, we can divide both sides of the equation by 2 to solve for x. This can be done as follows:

\frac{2x}{2} = \frac{12}{2}

x = 6

Conclusion

In this article, we solved the linear equation $2x - 5 = 7$ using basic algebraic techniques. We added 5 to both sides of the equation to eliminate the negative term, and then divided both sides by 2 to solve for x. The solution to the equation is x = 6.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is solved correctly.
When adding or subtracting terms, make sure to add or subtract the same value to both sides of the equation.
When dividing terms, make sure to divide both sides of the equation by the same value.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future economic trends.

Common Mistakes

Failing to follow the order of operations (PEMDAS) when solving linear equations.
Adding or subtracting different values to both sides of the equation.
Dividing both sides of the equation by a value that is not a factor of the term with the variable.

Final Thoughts

Solving linear equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), add or subtract the same value to both sides of the equation, and divide both sides of the equation by the same value. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Additional Resources

Khan Academy: Linear Equations
Mathway: Linear Equations
Wolfram Alpha: Linear Equations

Frequently Asked Questions

Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable is 1.
Q: How do I solve a linear equation?
A: To solve a linear equation, follow the steps outlined in this article: add or subtract the same value to both sides of the equation, and divide both sides of the equation by the same value.
Q: What are some real-world applications of linear equations?
A: Linear equations have numerous real-world applications, including physics, engineering, and economics.

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Introduction

In the previous article, we discussed how to solve the linear equation $2x - 5 = 7$. However, we understand that there may be many questions and doubts that readers may have about solving linear equations. In this article, we will address 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 is 1. In other words, a linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are constants.

Q: How do I solve a linear equation?

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

Add or subtract the same value to both sides of the equation to isolate the term with the variable.
Divide both sides of the equation by the same value to solve for the variable.

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

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

Failing to follow the order of operations (PEMDAS) when solving linear equations.
Adding or subtracting different values to both sides of the equation.
Dividing both sides of the equation by a value that is not a factor of the term with the variable.

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

A: To determine if an equation is linear or not, look for the highest power of the variable. If the highest power of the variable is 1, then the equation is linear. If the highest power of the variable is greater than 1, then the equation is not linear.

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

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the steps involved in solving linear equations, as using a calculator without understanding the underlying math can lead to errors.

Q: How do I check my answer when solving a linear equation?

A: To check your answer when solving a linear equation, plug the solution back into the original equation and verify that it is true. If the solution satisfies the original equation, then it is the correct solution.

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

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

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future economic trends.

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

A: Yes, you can use linear equations to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously. To solve a system of equations, use the substitution or elimination method to find the solution.

Q: How do I graph a linear equation?

A: To graph a linear equation, use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept. Plot the y-intercept on the graph, and then use the slope to find the other points on the line.

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

A: No, you cannot use linear equations to solve quadratic equations. Quadratic equations are equations in which the highest power of the variable is 2, and they require a different set of techniques to solve.

Q: How do I use linear equations to model real-world problems?

A: To use linear equations to model real-world problems, identify the variables and constants in the problem, and then write an equation that represents the relationship between the variables. Use the equation to make predictions and solve for the unknown variables.

Conclusion

In this article, we addressed some of the most frequently asked questions about solving linear equations. We hope that this article has provided you with a better understanding of linear equations and how to solve them. Remember to follow the steps outlined in this article, and don't hesitate to ask for help if you need it.

Additional Resources

Khan Academy: Linear Equations
Mathway: Linear Equations
Wolfram Alpha: Linear Equations