Solve The Equation: $\[ 4x + 4 - 2(x + 1) = 9x + 8 \\]The Solution Set Is \[$\square\$\]. (Simplify Your Answer.)
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 involves simplifying and isolating the variable. We will use the given equation as an example and walk through the steps to find the solution set.
The Given Equation
The equation we will be solving is:
4x + 4 - 2(x + 1) = 9x + 8
Step 1: Simplify the Left Side of the Equation
To simplify the left side of the equation, we need to distribute the -2 to the terms inside the parentheses.
4x + 4 - 2x - 2 = 9x + 8
Now, let's combine like terms:
2x + 2 = 9x + 8
Step 2: Isolate the Variable
Our goal is to isolate the variable x. To do this, we need to get all the x terms on one side of the equation and the constants on the other side.
Let's subtract 2x from both sides of the equation:
2 = 7x + 8
Step 3: Simplify the Right Side of the Equation
Now, let's simplify the right side of the equation by subtracting 8 from both sides:
-6 = 7x
Step 4: Solve for x
To solve for x, we need to divide both sides of the equation by 7:
x = -6/7
Conclusion
In this article, we solved a linear equation by simplifying and isolating the variable. We walked through the steps to find the solution set, which is x = -6/7. This equation is a great example of how to simplify and solve linear equations, and we hope this article has provided a clear and concise guide for students to follow.
Tips and Tricks
- When simplifying the left side of the equation, make sure to distribute the coefficients to all the terms inside the parentheses.
- When isolating the variable, try to get all the x terms on one side of the equation and the constants on the other side.
- When solving for x, make sure to divide both sides of the equation by the coefficient of x.
Common Mistakes to Avoid
- Not distributing the coefficients to all the terms inside the parentheses.
- Not combining like terms.
- Not isolating the variable correctly.
Real-World Applications
Linear equations have many real-world applications, such as:
- Modeling population growth
- Calculating interest rates
- Determining the cost of goods
Practice Problems
Try solving the following linear equations:
- 2x + 5 = 11
- x - 3 = 7
- 4x - 2 = 12
Conclusion
Introduction
In our previous article, we walked through the steps to solve a linear equation. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts or questions you may have. In this article, we will address some common questions and provide examples to help you better understand how to solve linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I simplify the left side of a linear equation?
A: To simplify the left side of a linear equation, you need to distribute the coefficients to all the terms inside the parentheses. For example, if you have the equation 2(x + 3), you would distribute the 2 to both the x and the 3, resulting in 2x + 6.
Q: How do I isolate the variable in a linear equation?
A: To isolate the variable in a linear equation, you need to get all the x terms on one side of the equation and the constants on the other side. You can do this by adding or subtracting the same value to both sides of the equation.
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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation x + 2 = 5 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I solve a linear equation with fractions?
A: To solve a linear equation with fractions, you need to follow the same steps as you would with a linear equation without fractions. However, you may need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
Q: What is the order of operations when solving a linear equation?
A: The order of operations when solving a linear equation is:
- Parentheses: Evaluate any expressions inside parentheses.
- Exponents: Evaluate any exponential expressions.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.
Q: Can I use a calculator to solve a linear equation?
A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your work by plugging the solution back into the original equation to make sure it's true.
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 coefficients to all the terms inside the parentheses.
- Not combining like terms.
- Not isolating the variable correctly.
- Not checking your work by plugging the solution back into the original equation.
Conclusion
Solving linear equations can be a challenging task, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and avoiding common mistakes, you can confidently solve linear equations and apply them to real-world problems.
Practice Problems
Try solving the following linear equations:
- 2x + 5 = 11
- x - 3 = 7
- 4x - 2 = 12
Additional Resources
For more practice problems and examples, check out the following resources:
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- IXL: Linear Equations
Conclusion
We hope this Q&A guide has helped you better understand how to solve linear equations. Remember to practice regularly and avoid common mistakes to become proficient in solving linear equations.