Solve The Equation: ${ 7x - 2(2x + 7) + 4x = 42 }$
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, to help you understand the process and build your confidence in tackling similar problems.
The Equation
The equation we will be solving is:
7x - 2(2x + 7) + 4x = 42
This equation appears to be complex, but don't worry, we will break it down into manageable steps.
Step 1: Distribute the Negative Sign
The first step in solving this equation is to distribute the negative sign to the terms inside the parentheses. This will help us simplify the equation and make it easier to work with.
7x - 2(2x + 7) + 4x = 42
Distributing the negative sign, we get:
7x - 4x - 14 + 4x = 42
Step 2: Combine Like Terms
Now that we have distributed the negative sign, we can combine like terms. In this case, we have three terms with the variable x: 7x, -4x, and 4x. We can combine these terms by adding or subtracting their coefficients.
7x - 4x - 14 + 4x = 42
Combining like terms, we get:
7x - 14 = 42
Step 3: Add 14 to Both Sides
The next step is to add 14 to both sides of the equation. This will help us isolate the variable x and solve for its value.
7x - 14 = 42
Adding 14 to both sides, we get:
7x = 56
Step 4: Divide Both Sides by 7
Finally, we can solve for x by dividing both sides of the equation by 7.
7x = 56
Dividing both sides by 7, we get:
x = 8
Conclusion
And there you have it! We have solved the equation 7x - 2(2x + 7) + 4x = 42 step by step. By following these simple steps, you can solve similar linear equations and build your confidence in mathematics.
Tips and Tricks
Here are some tips and tricks to help you solve linear equations:
- Distribute the negative sign: When you see a negative sign in front of a set of parentheses, distribute it to the terms inside the parentheses.
- Combine like terms: When you have multiple terms with the same variable, combine them by adding or subtracting their coefficients.
- Add or subtract the same value to both sides: When you want to isolate a variable, add or subtract the same value to both sides of the equation.
- Divide both sides by a coefficient: When you want to solve for a variable, divide both sides of the equation by its coefficient.
Practice Problems
Here are some practice problems to help you reinforce your understanding of solving linear equations:
- 2x + 5 = 11
- 3x - 2 = 7
- x + 2 = 9
Real-World Applications
Linear equations have many real-world applications, including:
- Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Conclusion
Introduction
In our previous article, we discussed how to solve linear equations step by step. 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 answers to help you better understand solving 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. In other words, it's an equation that can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I know if an equation is linear or not?
A: To determine if an equation is linear, look for the highest power of the variable. If it's 1, then the equation is linear. If it's greater than 1, then the equation is not linear.
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, 2x + 3 = 5 is a linear equation, while 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, follow these steps:
- Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
- Simplify the equation by canceling out any common factors.
- Solve for the variable using the steps outlined in our previous article.
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 the equation, so you can verify the answer and ensure that it's correct.
Q: What if I have a linear equation with multiple variables?
A: If you have a linear equation with multiple variables, you can solve it using the same steps outlined in our previous article. However, you may need to use substitution or elimination methods to solve for the variables.
Q: Can I use linear equations to solve real-world problems?
A: Yes, linear equations can be used to solve real-world problems. For example, you can use linear equations to calculate interest rates, investment returns, and other financial metrics. You can also use linear equations to model population growth, chemical reactions, and other scientific phenomena.
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 negative sign to the terms inside the parentheses.
- Not combining like terms.
- Not adding or subtracting the same value to both sides of the equation.
- Not dividing both sides by a coefficient.
Conclusion
Solving linear equations is a fundamental skill that has many real-world applications. By following the steps outlined in this article and practicing with different types of equations, you can become proficient in solving linear equations and tackle more complex problems. Remember to distribute the negative sign, combine like terms, add or subtract the same value to both sides, and divide both sides by a coefficient. With practice and patience, you can become a master of solving linear equations.
Practice Problems
Here are some practice problems to help you reinforce your understanding of solving linear equations:
- 2x + 3 = 7
- 3x - 2 = 5
- x + 2 = 9
Real-World Applications
Linear equations have many real-world applications, including:
- Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Conclusion
Solving linear equations is a fundamental skill that has many real-world applications. By following the steps outlined in this article and practicing with different types of equations, you can become proficient in solving linear equations and tackle more complex problems. Remember to distribute the negative sign, combine like terms, add or subtract the same value to both sides, and divide both sides by a coefficient. With practice and patience, you can become a master of solving linear equations.