Solve $8 = 2(10 + 3c)$.
Introduction
In this article, we will delve into solving a linear equation, specifically the equation 8 = 2(10 + 3c). Linear equations are a fundamental concept in mathematics, and solving them is crucial for various applications in science, engineering, and economics. The equation 8 = 2(10 + 3c) is a simple yet effective example of a linear equation that requires careful analysis and step-by-step solution.
Understanding the Equation
The given equation is 8 = 2(10 + 3c). To solve this equation, we need to isolate the variable c. The equation is a linear equation because it can be written in the form ax + b = c, where a, b, and c are constants. In this case, the equation can be rewritten as 8 = 20 + 6c.
Step 1: Distribute the Coefficient
To solve the equation, we need to distribute the coefficient 2 to the terms inside the parentheses. This means multiplying 2 by 10 and 2 by 3c. The equation becomes 8 = 20 + 6c.
Step 2: Isolate the Variable
To isolate the variable c, we need to get rid of the constant term 20 on the right-hand side of the equation. We can do this by subtracting 20 from both sides of the equation. This gives us -12 = 6c.
Step 3: Solve for c
Now that we have isolated the variable c, we can solve for its value. To do this, we need to divide both sides of the equation by 6. This gives us c = -12/6.
Step 4: Simplify the Expression
To simplify the expression -12/6, we can divide both the numerator and the denominator by their greatest common divisor, which is 6. This gives us c = -2.
Conclusion
In conclusion, we have solved the linear equation 8 = 2(10 + 3c) by following the steps outlined above. We distributed the coefficient, isolated the variable, and solved for its value. The final solution is c = -2.
Example Use Case
The equation 8 = 2(10 + 3c) can be used to model real-world scenarios, such as calculating the cost of producing a certain number of items. For example, if a company produces 10 items and each item costs $3 to produce, the total cost of producing 10 items is $30. However, if the company produces 10 + 3c items, the total cost of producing these items is 2(30 + 3c). By solving the equation 8 = 2(10 + 3c), we can find the value of c, which represents the number of items produced.
Tips and Tricks
When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to isolate the variable by performing the necessary operations. Additionally, it's crucial to simplify the expression and to check the solution by plugging it back into the original equation.
Common Mistakes
When solving linear equations, some common mistakes include:
- Not distributing the coefficient correctly
- Not isolating the variable correctly
- Not simplifying the expression correctly
- Not checking the solution by plugging it back into the original equation
Real-World Applications
Linear equations have numerous real-world applications, including:
- Calculating the cost of producing a certain number of items
- Modeling population growth
- Calculating the interest on a loan
- Determining the amount of fuel needed for a trip
Conclusion
In conclusion, solving linear equations is a crucial skill that has numerous real-world applications. By following the steps outlined above and by being aware of common mistakes, we can solve linear equations with ease and confidence. Whether it's calculating the cost of producing a certain number of items or modeling population growth, linear equations are an essential tool for problem-solving in various fields.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that requires careful analysis and step-by-step solution. By following the steps outlined above and by being aware of common mistakes, we can solve linear equations with ease and confidence. Whether it's calculating the cost of producing a certain number of items or modeling population growth, linear equations are an essential tool for problem-solving in various fields.
Introduction
In our previous article, we solved the linear equation 8 = 2(10 + 3c) by following the steps outlined above. However, we understand that solving linear equations can be a challenging task, and many readers may have questions about the process. In this article, we will address some of the most frequently asked questions about solving linear equations, specifically the equation 8 = 2(10 + 3c).
Q: What is the first step in solving a linear equation?
A: The first step in solving a linear equation is to distribute the coefficient to the terms inside the parentheses. This means multiplying the coefficient by each term inside the parentheses.
Q: How do I know if I have distributed the coefficient correctly?
A: To check if you have distributed the coefficient correctly, simply multiply the coefficient by each term inside the parentheses and see if the result is the same as the original equation.
Q: What is the next step after distributing the coefficient?
A: After distributing the coefficient, the next step is to isolate the variable by performing the necessary operations. This may involve adding or subtracting a constant term to one side of the equation.
Q: How do I isolate the variable?
A: To isolate the variable, you need to get rid of the constant term on the same side of the equation as the variable. You can do this by adding or subtracting a constant term to both sides of the equation.
Q: What is the final step in solving a linear equation?
A: The final step in solving a linear equation is to simplify the expression and check the solution by plugging it back into the original equation.
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 coefficient correctly
- Not isolating the variable correctly
- Not simplifying the expression correctly
- Not checking the solution by plugging it back into the original equation
Q: How do I check my solution?
A: To check your solution, simply plug the value of the variable back into the original equation and see if the result is true.
Q: What are some real-world applications of linear equations?
A: Linear equations have numerous real-world applications, including:
- Calculating the cost of producing a certain number of items
- Modeling population growth
- Calculating the interest on a loan
- Determining the amount of fuel needed for a trip
Q: Can I use linear equations to solve problems in other fields?
A: Yes, linear equations can be used to solve problems in other fields, including physics, engineering, and economics.
Q: How do I know if a problem can be solved using a linear equation?
A: If a problem involves a variable and a constant term, and the relationship between the variable and the constant term is linear, then the problem can be solved using a linear equation.
Q: What are some tips for solving linear equations?
A: Some tips for solving linear equations include:
- Following the order of operations (PEMDAS)
- Isolating the variable by performing the necessary operations
- Simplifying the expression
- Checking the solution by plugging it back into the original equation
Conclusion
In conclusion, solving linear equations is a crucial skill that has numerous real-world applications. By following the steps outlined above and by being aware of common mistakes, we can solve linear equations with ease and confidence. Whether it's calculating the cost of producing a certain number of items or modeling population growth, linear equations are an essential tool for problem-solving in various fields.
Final Thoughts
Solving linear equations is a fundamental concept in mathematics that requires careful analysis and step-by-step solution. By following the steps outlined above and by being aware of common mistakes, we can solve linear equations with ease and confidence. Whether it's calculating the cost of producing a certain number of items or modeling population growth, linear equations are an essential tool for problem-solving in various fields.