Solve For $y$.$6y + 6(y + 2) = 36$Simplify Your Answer As Much As Possible.$y = $[/tex]
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 equation $6y + 6(y + 2) = 36$ as an example to demonstrate the step-by-step process.
Understanding the Equation
Before we dive into solving the equation, let's break it down and understand what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, y) is 1. The equation is also a quadratic equation, as it involves a quadratic expression on the left-hand side.
Step 1: Simplify the Equation
The first step in solving the equation is to simplify it by combining like terms. In this case, we can combine the two terms involving y:
We can start by distributing the 6 to the terms inside the parentheses:
Now, we can combine the two like terms:
Step 2: Isolate the Variable
The next step is to isolate the variable y by getting rid of the constant term on the left-hand side. We can do this by subtracting 12 from both sides of the equation:
Step 3: Solve for y
Now that we have isolated the variable, we can solve for y by dividing both sides of the equation by 12:
Conclusion
In this article, we have demonstrated the step-by-step process of solving a linear equation. We started by simplifying the equation by combining like terms, then isolated the variable by getting rid of the constant term, and finally solved for y by dividing both sides of the equation by the coefficient of y. By following these steps, we were able to simplify the equation and find the value of y.
Tips and Tricks
Here are some tips and tricks to help you solve linear equations:
- Combine like terms: When simplifying an equation, combine like terms to make it easier to solve.
- Isolate the variable: Get rid of the constant term on the left-hand side by subtracting or adding it to both sides of the equation.
- Use inverse operations: Use inverse operations (such as addition and subtraction, or multiplication and division) to isolate the variable.
- Check your work: Always check your work by plugging the solution back into the original equation to make sure it is true.
Real-World Applications
Linear equations have many real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
- Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and computer networks.
- Economics: Linear equations are used to model economic systems, including supply and demand, cost and revenue, and profit and loss.
Conclusion
Introduction
In our previous article, we discussed the step-by-step process of solving linear equations. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help you better understand and solve linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (in this case, y) is 1. It is a simple equation that can be solved by using basic algebraic operations.
Q: How do I simplify a linear equation?
A: To simplify a linear equation, combine like terms by adding or subtracting the coefficients of the same variable. For example, in the equation $6y + 6(y + 2) = 36$, we can combine the two terms involving y by distributing the 6 to the terms inside the parentheses and then combining the like terms.
Q: How do I isolate the variable in a linear equation?
A: To isolate the variable, get rid of the constant term on the left-hand side by subtracting or adding it to both sides of the equation. For example, in the equation $12y = 24$, we can isolate the variable y by dividing both sides of the equation by 12.
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 $y = 2x + 3$ is a linear equation, while the equation $y = x^2 + 2x + 1$ is a quadratic equation.
Q: How do I solve a linear equation with fractions?
A: To solve a linear equation with fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions. For example, in the equation $\frac{y}{2} = 3$, we can multiply both sides of the equation by 2 to eliminate the fraction.
Q: Can I use a calculator to solve linear equations?
A: 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 to make sure it is 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 combining like terms: Failing to combine like terms can make the equation more difficult to solve.
- Not isolating the variable: Failing to isolate the variable can make it difficult to solve for the variable.
- Not checking your work: Failing to check your work can lead to incorrect solutions.
Q: How can I practice solving linear equations?
A: There are many ways to practice solving linear equations, including:
- Using online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and practice problems to help you improve your skills.
- Working with a tutor: Working with a tutor can provide one-on-one instruction and feedback to help you improve your skills.
- Solving problems on your own: Solving problems on your own can help you develop problem-solving skills and build confidence in your abilities.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and practicing with online resources, working with a tutor, or solving problems on your own, you can improve your skills and become proficient in solving linear equations. Remember to combine like terms, isolate the variable, and check your work to ensure that you are solving the equation correctly.