Producing a Pair of Like Terms with Opposite Coefficients

Oct 13, 2025 by windailys.com 60 views

Introduction

In algebra, when working with equations, it's often necessary to manipulate them to achieve a specific goal. One common task is to produce a pair of like terms with opposite coefficients. This can be achieved by multiplying the first equation by a specific value. In this article, we will explore the concept of multiplying the first equation to produce a pair of like terms with opposite coefficients.

Understanding Like Terms

Like terms are algebraic expressions that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1. When we multiply like terms, we add their coefficients. For instance, (2x + 4x) = 6x.

Producing a Pair of Like Terms with Opposite Coefficients

To produce a pair of like terms with opposite coefficients, we need to multiply the first equation by a value that will result in the desired coefficients. Let's consider an example to illustrate this concept.

Suppose we have the equation 2x = 6. To produce a pair of like terms with opposite coefficients, we need to multiply the first equation by a value that will result in the coefficients being opposite. In this case, we can multiply the first equation by 3/2.

Why Multiply by 3/2?

Multiplying the first equation by 3/2 will result in the coefficients being opposite. To see why, let's perform the multiplication:

(2x) × (3/2) = 3x

Now, we have a pair of like terms with opposite coefficients: 3x and -6.

Alternative Solutions

While multiplying the first equation by 3/2 is one way to produce a pair of like terms with opposite coefficients, there are other solutions as well. Let's consider the options provided in the discussion category:

A. $\frac{2}{3}$

Multiplying the first equation by 2/3 will result in the coefficients being opposite. To see why, let's perform the multiplication:

(2x) × (2/3) = 4/3x

Now, we have a pair of like terms with opposite coefficients: 4/3x and -6.

B. $\frac{4}{3}$

Multiplying the first equation by 4/3 will result in the coefficients being opposite. To see why, let's perform the multiplication:

(2x) × (4/3) = 8/3x

Now, we have a pair of like terms with opposite coefficients: 8/3x and -6.

C. 6

Multiplying the first equation by 6 will result in the coefficients being opposite. To see why, let's perform the multiplication:

(2x) × 6 = 12x

Now, we have a pair of like terms with opposite coefficients: 12x and -6.

Conclusion

In conclusion, to produce a pair of like terms with opposite coefficients, we can multiply the first equation by a value that will result in the desired coefficients. In this article, we explored the concept of multiplying the first equation to produce a pair of like terms with opposite coefficients. We also considered alternative solutions and provided examples to illustrate the concept.

Key Takeaways

To produce a pair of like terms with opposite coefficients, we can multiply the first equation by a value that will result in the desired coefficients.
Multiplying the first equation by 3/2 will result in the coefficients being opposite.
Alternative solutions include multiplying the first equation by 2/3, 4/3, or 6.
When multiplying like terms, we add their coefficients.

Final Thoughts

Q: What are like terms?

A: Like terms are algebraic expressions that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: Why do we need to produce a pair of like terms with opposite coefficients?

A: Producing a pair of like terms with opposite coefficients is often necessary when working with equations. It allows us to simplify the equation and make it easier to solve.

Q: How do I multiply the first equation to produce a pair of like terms with opposite coefficients?

A: To multiply the first equation to produce a pair of like terms with opposite coefficients, you need to multiply the first equation by a value that will result in the desired coefficients. This value is often a fraction or a whole number.

Q: What are some common values that I can multiply the first equation by to produce a pair of like terms with opposite coefficients?

A: Some common values that you can multiply the first equation by to produce a pair of like terms with opposite coefficients include:

Q: How do I know which value to multiply the first equation by?

A: To determine which value to multiply the first equation by, you need to consider the coefficients of the like terms. You want to multiply the first equation by a value that will result in the coefficients being opposite.

Q: What if I multiply the first equation by a value that doesn't result in opposite coefficients?

A: If you multiply the first equation by a value that doesn't result in opposite coefficients, you may need to try a different value. You can also try multiplying the first equation by a different value or using a different method to solve the equation.

Q: Can I use this method to solve any equation?

A: This method can be used to solve equations that have like terms with opposite coefficients. However, it may not be the best method for all equations. You should consider the specific equation and the coefficients of the like terms before deciding which method to use.

Q: Are there any other methods that I can use to solve equations with like terms?

A: Yes, there are other methods that you can use to solve equations with like terms. Some common methods include:

Adding or subtracting like terms
Multiplying or dividing like terms
Using algebraic identities to simplify the equation

Q: How do I know which method to use?

A: To determine which method to use, you need to consider the specific equation and the coefficients of the like terms. You should also consider the level of difficulty of the equation and the time it will take to solve it.

Q: Can I use this method to solve equations with variables on both sides?

A: This method can be used to solve equations with variables on both sides. However, you may need to use additional steps to isolate the variable and solve the equation.

Q: Are there any tips or tricks that I can use to make this method easier?

A: Yes, there are several tips and tricks that you can use to make this method easier. Some common tips and tricks include:

Using a calculator to simplify the equation
Breaking down the equation into smaller parts
Using algebraic identities to simplify the equation
Checking your work to ensure that the equation is correct

Q: Can I use this method to solve equations with fractions?

A: This method can be used to solve equations with fractions. However, you may need to use additional steps to simplify the fraction and solve the equation.

Q: Are there any common mistakes that I should avoid when using this method?

A: Yes, there are several common mistakes that you should avoid when using this method. Some common mistakes include:

Multiplying the first equation by a value that doesn't result in opposite coefficients
Failing to simplify the equation
Failing to check your work
Using the wrong method to solve the equation

Q: Can I use this method to solve equations with decimals?

A: This method can be used to solve equations with decimals. However, you may need to use additional steps to simplify the decimal and solve the equation.

Q: Are there any other resources that I can use to learn more about this method?

A: Yes, there are several other resources that you can use to learn more about this method. Some common resources include:

Algebra textbooks
Online tutorials
Video lectures
Practice problems

Conclusion

In conclusion, producing a pair of like terms with opposite coefficients is an essential skill in algebra. By understanding how to multiply the first equation to achieve this goal, you can solve a wide range of problems and equations. Whether you're working with simple equations or complex algebraic expressions, this skill will serve you well.