Solving Multistep Equations With Variables On Both SidesExamine The First Two Steps Used To Solve The Equation:$\[ \frac{3}{4}(20y - 8) + 5 = \frac{1}{2}y + \frac{1}{4}(20y + 8) \\]$\[ 15y - 6 + 5 = \frac{1}{2}y + 5y +

by ADMIN 219 views

Introduction

Solving multistep equations with variables on both sides can be a challenging task for many students. These types of equations require a step-by-step approach to isolate the variable and find its value. In this article, we will examine the first two steps used to solve the equation 34(20y−8)+5=12y+14(20y+8)\frac{3}{4}(20y - 8) + 5 = \frac{1}{2}y + \frac{1}{4}(20y + 8). We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Distribute the Coefficients

The first step in solving the equation is to distribute the coefficients to the terms inside the parentheses. This will allow us to simplify the equation and make it easier to work with.

34(20y−8)+5=12y+14(20y+8)\frac{3}{4}(20y - 8) + 5 = \frac{1}{2}y + \frac{1}{4}(20y + 8)

To distribute the coefficients, we multiply each term inside the parentheses by the coefficient outside the parentheses.

34(20y)−34(8)+5=12y+14(20y)+14(8)\frac{3}{4}(20y) - \frac{3}{4}(8) + 5 = \frac{1}{2}y + \frac{1}{4}(20y) + \frac{1}{4}(8)

Simplifying the equation, we get:

15y−6+5=12y+5y+215y - 6 + 5 = \frac{1}{2}y + 5y + 2

Step 2: Combine Like Terms

The next step is to combine like terms on both sides of the equation. This will help us simplify the equation further and make it easier to solve.

15y−6+5=12y+5y+215y - 6 + 5 = \frac{1}{2}y + 5y + 2

Combining like terms on the left side of the equation, we get:

15y−1=12y+5y+215y - 1 = \frac{1}{2}y + 5y + 2

Combining like terms on the right side of the equation, we get:

15y−1=72y+215y - 1 = \frac{7}{2}y + 2

Why is it Important to Follow the Order of Operations?

Following the order of operations is crucial when solving multistep equations with variables on both sides. The order of operations states that we should perform operations in the following order:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following the order of operations, we can ensure that we are solving the equation correctly and avoiding any potential errors.

Tips and Tricks for Solving Multistep Equations

Here are some tips and tricks for solving multistep equations with variables on both sides:

  • Read the equation carefully: Before starting to solve the equation, read it carefully to make sure you understand what is being asked.
  • Use a step-by-step approach: Break down the solution into manageable steps and follow the order of operations.
  • Combine like terms: Combine like terms on both sides of the equation to simplify it further.
  • Check your work: Finally, check your work by plugging the solution back into the original equation to make sure it is true.

Conclusion

Solving multistep equations with variables on both sides requires a step-by-step approach and a clear understanding of the order of operations. By following the steps outlined in this article, you can solve these types of equations with confidence. Remember to read the equation carefully, use a step-by-step approach, combine like terms, and check your work. With practice and patience, you will become proficient in solving multistep equations with variables on both sides.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving multistep equations with variables on both sides:

  • Not following the order of operations: Failing to follow the order of operations can lead to errors and incorrect solutions.
  • Not combining like terms: Failing to combine like terms can make the equation more difficult to solve and increase the risk of errors.
  • Not checking your work: Failing to check your work can lead to incorrect solutions and a lack of confidence in your abilities.

Real-World Applications

Solving multistep equations with variables on both sides has many real-world applications. Here are a few examples:

  • Science and Engineering: Multistep equations are used to model real-world phenomena, such as the motion of objects, the behavior of electrical circuits, and the growth of populations.
  • Finance: Multistep equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Computer Science: Multistep equations are used to solve problems in computer science, such as graph theory, network flow, and optimization.

Final Thoughts

Introduction

Solving multistep equations with variables on both sides can be a challenging task for many students. In our previous article, we examined the first two steps used to solve the equation 34(20y−8)+5=12y+14(20y+8)\frac{3}{4}(20y - 8) + 5 = \frac{1}{2}y + \frac{1}{4}(20y + 8). In this article, we will provide a Q&A section to help you better understand the concept and provide additional tips and tricks for solving multistep equations with variables on both sides.

Q: What is a multistep equation?

A: A multistep equation is an equation that requires multiple steps to solve. It typically involves variables on both sides of the equation and may require the use of inverse operations, such as addition and subtraction, multiplication and division, and exponentiation.

Q: Why is it important to follow the order of operations when solving multistep equations?

A: Following the order of operations is crucial when solving multistep equations because it ensures that we are performing the operations in the correct order. This helps to avoid errors and ensures that we are solving the equation correctly.

Q: What are some common mistakes to avoid when solving multistep equations?

A: Some common mistakes to avoid when solving multistep equations include:

  • Not following the order of operations
  • Not combining like terms
  • Not checking your work
  • Not using inverse operations correctly

Q: How do I know when to use inverse operations when solving multistep equations?

A: Inverse operations are used to "undo" the effects of an operation. For example, if an equation involves addition, we may need to use subtraction to "undo" the addition. Similarly, if an equation involves multiplication, we may need to use division to "undo" the multiplication.

Q: What are some tips and tricks for solving multistep equations?

A: Here are some tips and tricks for solving multistep equations:

  • Read the equation carefully before starting to solve it
  • Use a step-by-step approach to break down the solution into manageable steps
  • Combine like terms on both sides of the equation to simplify it further
  • Check your work by plugging the solution back into the original equation to make sure it is true
  • Use inverse operations correctly to "undo" the effects of an operation

Q: How do I know if I have solved the equation correctly?

A: To check if you have solved the equation correctly, plug the solution back into the original equation and make sure it is true. If the solution is true, then you have solved the equation correctly.

Q: What are some real-world applications of solving multistep equations?

A: Solving multistep equations has many real-world applications, including:

  • Science and engineering: Multistep equations are used to model real-world phenomena, such as the motion of objects, the behavior of electrical circuits, and the growth of populations.
  • Finance: Multistep equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Computer science: Multistep equations are used to solve problems in computer science, such as graph theory, network flow, and optimization.

Conclusion

Solving multistep equations with variables on both sides requires a step-by-step approach and a clear understanding of the order of operations. By following the steps outlined in this article and using the tips and tricks provided, you can solve these types of equations with confidence. Remember to read the equation carefully, use a step-by-step approach, combine like terms, and check your work. With practice and patience, you will become proficient in solving multistep equations with variables on both sides.

Additional Resources

For additional resources and practice problems, check out the following websites:

Final Thoughts

Solving multistep equations with variables on both sides requires a step-by-step approach and a clear understanding of the order of operations. By following the steps outlined in this article and using the tips and tricks provided, you can solve these types of equations with confidence. Remember to read the equation carefully, use a step-by-step approach, combine like terms, and check your work. With practice and patience, you will become proficient in solving multistep equations with variables on both sides.