Solve The Equation:${ 3x + 1 - 5 = -8 + 6x - 8 }$
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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, step by step, and provide a clear understanding of the process involved.
What is a Linear Equation?
A linear equation is an 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.
The Given Equation
The equation we will be solving is:
3x + 1 - 5 = -8 + 6x - 8
Step 1: Simplify the Equation
To simplify the equation, we need to combine like terms on both sides of the equation.
3x + 1 - 5 = -8 + 6x - 8
First, let's simplify the left-hand side of the equation:
3x - 4 = -8 + 6x - 8
Next, let's simplify the right-hand side of the equation:
-8 + 6x - 8 = -16 + 6x
So, the simplified equation is:
3x - 4 = -16 + 6x
Step 2: Isolate the Variable
To isolate the variable x, we need to get all the terms with x on one side of the equation and the constant terms on the other side.
Let's start by adding 4 to both sides of the equation:
3x - 4 + 4 = -16 + 6x + 4
This simplifies to:
3x = -12 + 6x
Step 3: Get Rid of the Constant Term
To get rid of the constant term -12, we need to add 12 to both sides of the equation:
3x + 12 = -12 + 6x + 12
This simplifies to:
3x + 12 = 6x
Step 4: Subtract the Variable Term
To subtract the variable term 3x from both sides of the equation, we need to subtract 3x from both sides:
3x - 3x + 12 = 6x - 3x
This simplifies to:
12 = 3x
Step 5: Solve for x
To solve for x, we need to divide both sides of the equation by 3:
12/3 = 3x/3
This simplifies to:
4 = x
Conclusion
In this article, we solved a linear equation step by step, using basic algebraic operations. We started by simplifying the equation, then isolated the variable, got rid of the constant term, subtracted the variable term, and finally solved for x. The final solution is x = 4.
Tips and Tricks
- When solving linear equations, always start by simplifying the equation.
- Use basic algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable.
- Get rid of constant terms by adding or subtracting the same value from both sides of the equation.
- Subtract variable terms by subtracting the same value from both sides of the equation.
- Finally, solve for x by dividing both sides of the equation by the coefficient of x.
Practice Problems
Try solving the following linear equations:
- 2x + 3 = 5 - x
- x - 2 = 3x + 1
- 4x + 2 = 2x - 3
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 trends.
Conclusion
In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve linear equations with ease. Remember to simplify the equation, isolate the variable, get rid of constant terms, subtract variable terms, and finally solve for x. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.
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Introduction
In our previous article, we discussed the steps involved in solving linear equations. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide to help you understand and solve linear equations.
Q1: What is a linear equation?
A linear equation is an 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.
Q2: How do I simplify a linear equation?
To simplify a linear equation, you need to combine like terms on both sides of the equation. This involves adding or subtracting the same value from both sides of the equation.
Q3: How do I isolate the variable in a linear equation?
To isolate the variable, you need to get all the terms with the variable on one side of the equation and the constant terms on the other side. This can be done by adding or subtracting the same value from both sides of the equation.
Q4: What is the difference between a linear equation and a quadratic equation?
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.
Q5: How do I solve a linear equation with fractions?
To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
Q6: Can I use a calculator to solve linear equations?
Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by plugging the solution back into the original equation.
Q7: How do I check my solution to a linear equation?
To check your solution, you need to plug the solution back into the original equation and verify that it's true.
Q8: What are some common mistakes to avoid when solving linear equations?
Some common mistakes to avoid when solving linear equations include:
- Not simplifying the equation before solving
- Not isolating the variable
- Not checking the solution
- Not using the correct order of operations
Q9: Can I use linear equations to solve real-world problems?
Yes, linear equations can be used to solve real-world problems. For example, you can use linear equations to model the motion of objects under constant acceleration, design and optimize systems, and make predictions about future trends.
Q10: How can I practice solving linear equations?
You can practice solving linear equations by working through example problems, using online resources, and taking practice quizzes.
Conclusion
In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and practicing regularly, 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
Practice Problems
Try solving the following linear equations:
- 2x + 3 = 5 - x
- x - 2 = 3x + 1
- 4x + 2 = 2x - 3
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 trends.