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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 isolating the variable w. We will use the equation 6(w - 3) = 36 as an example and walk through the step-by-step process of solving for w.
Understanding the Equation
The given equation is 6(w - 3) = 36. To solve for w, we need to isolate the variable w on one side of the equation. The equation involves a coefficient of 6, which is multiplied by the expression (w - 3). Our goal is to simplify the equation and isolate w.
Step 1: Distribute the Coefficient
The first step in solving the equation is to distribute the coefficient 6 to the terms inside the parentheses. This means multiplying 6 by each term inside the parentheses.
6(w - 3) = 6w - 18
By distributing the coefficient, we have simplified the equation and made it easier to work with.
Step 2: Add or Subtract Constants
The next step is to add or subtract constants to isolate the variable w. In this case, we need to add 18 to both sides of the equation to get rid of the negative term.
6w - 18 + 18 = 36 + 18
By adding 18 to both sides, we have eliminated the negative term and simplified the equation further.
Step 3: Simplify the Equation
Now that we have added 18 to both sides, we can simplify the equation by combining like terms.
6w = 54
By combining like terms, we have simplified the equation and isolated the variable w.
Step 4: Divide by the Coefficient
The final step is to divide both sides of the equation by the coefficient 6 to solve for w.
w = 54/6
By dividing both sides by 6, we have isolated the variable w and solved the equation.
Conclusion
Solving linear equations is a crucial skill for students to master, and it requires a step-by-step approach. By following the steps outlined in this article, we have solved the equation 6(w - 3) = 36 and isolated the variable w. The final solution is w = 9.
Tips and Tricks
- Always start by simplifying the equation and isolating the variable.
- Use the distributive property to multiply coefficients to terms inside parentheses.
- Add or subtract constants to eliminate negative terms.
- Simplify the equation by combining like terms.
- Divide both sides of the equation by the coefficient to solve for the variable.
Real-World Applications
Solving linear equations has numerous real-world applications, including:
- Physics: Solving linear equations is essential in physics to describe the motion of objects.
- Engineering: Linear equations are used to design and optimize systems.
- Economics: Linear equations are used to model economic systems and make predictions.
Common Mistakes
- Failing to simplify the equation and isolate the variable.
- Not using the distributive property to multiply coefficients.
- Not adding or subtracting constants to eliminate negative terms.
- Not simplifying the equation by combining like terms.
Conclusion
Introduction
In our previous article, we walked through the step-by-step process of solving a linear equation, 6(w - 3) = 36. We covered the basics of solving linear equations and provided tips and tricks for success. In this article, we will answer 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, it is 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?
A: To determine if an equation is linear, look for the highest power of the variable. If the highest power is 1, then the equation is linear. For example, the equation 2x + 3 = 5 is linear because the highest power of x is 1.
Q: What is the distributive property?
A: The distributive property is a mathematical property that allows us to multiply a coefficient to a term inside parentheses. For example, in the equation 6(w - 3), we can use the distributive property to multiply 6 to the terms inside the parentheses: 6w - 18.
Q: How do I add or subtract constants?
A: To add or subtract constants, simply add or subtract the numbers on both sides of the equation. For example, in the equation 6w - 18 + 18 = 36 + 18, we can add 18 to both sides to eliminate the negative term.
Q: What is the difference between a coefficient and a constant?
A: A coefficient is a number that is multiplied to a variable, while a constant is a number that is not multiplied to a variable. For example, in the equation 2x + 3, the 2 is a coefficient and the 3 is a constant.
Q: How do I simplify an equation?
A: To simplify an equation, combine like terms and eliminate any negative terms. For example, in the equation 6w - 18 + 18 = 36 + 18, we can simplify the equation by combining like terms: 6w = 54.
Q: What is the final step in solving a linear equation?
A: The final step in solving a linear equation is to divide both sides of the equation by the coefficient to solve for the variable. For example, in the equation 6w = 54, we can divide both sides by 6 to solve for w: w = 9.
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 simplify the equation and isolate the variable.
- Not using the distributive property to multiply coefficients.
- Not adding or subtracting constants to eliminate negative terms.
- Not simplifying the equation by combining like terms.
Q: How do I apply linear equations to real-world problems?
A: Linear equations have numerous real-world applications, including:
- Physics: Solving linear equations is essential in physics to describe the motion of objects.
- Engineering: Linear equations are used to design and optimize systems.
- Economics: Linear equations are used to model economic systems and make predictions.
Conclusion
Solving linear equations is a fundamental concept in mathematics, and it requires a step-by-step approach. By following the steps outlined in this article, we have answered some of the most frequently asked questions about solving linear equations. Remember to always simplify the equation and isolate the variable, use the distributive property, add or subtract constants, simplify the equation, and divide both sides by the coefficient to solve for the variable.