Examine The Two-step Equation.${ -\frac{7}{4}+\frac{x}{4}=2 }$Which Property Of Operations Allows You To Add The Same Constant Term To Both Sides?A. Multiplication Property Of Equality B. Division Property Of Equality C. Addition Property

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Introduction

In mathematics, equations are a fundamental concept that help us solve problems and understand relationships between variables. A two-step equation is a type of equation that requires two steps to solve, involving addition, subtraction, multiplication, or division operations. In this article, we will examine a two-step equation and explore the property of operations that allows us to add the same constant term to both sides.

The Two-Step Equation

The given two-step equation is:

−74+x4=2{ -\frac{7}{4}+\frac{x}{4}=2 }

To solve this equation, we need to isolate the variable x. However, the equation is not in a simple form, and we need to use a property of operations to make it easier to solve.

Adding the Same Constant Term to Both Sides

The property of operations that allows us to add the same constant term to both sides of an equation is the Addition Property of Equality. This property states that if we add the same value to both sides of an equation, the equation remains true.

Why is the Addition Property of Equality Important?

The Addition Property of Equality is a fundamental concept in mathematics that helps us solve equations and inequalities. It allows us to simplify equations by adding or subtracting the same value from both sides, making it easier to isolate the variable.

How to Use the Addition Property of Equality

To use the Addition Property of Equality, we need to add the same value to both sides of the equation. In this case, we can add 7/4 to both sides of the equation:

−74+x4+74=2+74{ -\frac{7}{4}+\frac{x}{4}+\frac{7}{4}=2+\frac{7}{4} }

This simplifies the equation to:

x4=154{ \frac{x}{4}=\frac{15}{4} }

Solving for x

Now that we have simplified the equation, we can solve for x by multiplying both sides of the equation by 4:

x=15{ x=15 }

Conclusion

In conclusion, the two-step equation −74+x4=2{ -\frac{7}{4}+\frac{x}{4}=2 } requires the use of the Addition Property of Equality to add the same constant term to both sides. This property allows us to simplify the equation and isolate the variable x. By understanding and applying the Addition Property of Equality, we can solve equations and inequalities with ease.

Key Takeaways

  • The Addition Property of Equality states that if we add the same value to both sides of an equation, the equation remains true.
  • The Addition Property of Equality is a fundamental concept in mathematics that helps us solve equations and inequalities.
  • To use the Addition Property of Equality, we need to add the same value to both sides of the equation.
  • By applying the Addition Property of Equality, we can simplify equations and isolate the variable.

Real-World Applications

The Addition Property of Equality has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the Addition Property of Equality is used to calculate the total energy of a system, while in engineering, it is used to design and optimize systems. In economics, the Addition Property of Equality is used to calculate the total cost of a product or service.

Common Mistakes to Avoid

When using the Addition Property of Equality, it is essential to avoid common mistakes such as:

  • Adding different values to both sides of the equation
  • Subtracting the same value from both sides of the equation
  • Not simplifying the equation after adding or subtracting the same value

By avoiding these common mistakes, we can ensure that our solutions are accurate and reliable.

Final Thoughts

Q: What is a two-step equation?

A: A two-step equation is a type of equation that requires two steps to solve, involving addition, subtraction, multiplication, or division operations.

Q: What is the Addition Property of Equality?

A: The Addition Property of Equality states that if we add the same value to both sides of an equation, the equation remains true.

Q: Why is the Addition Property of Equality important?

A: The Addition Property of Equality is a fundamental concept in mathematics that helps us solve equations and inequalities. It allows us to simplify equations by adding or subtracting the same value from both sides, making it easier to isolate the variable.

Q: How do I use the Addition Property of Equality?

A: To use the Addition Property of Equality, you need to add the same value to both sides of the equation. For example, if you have the equation x + 3 = 7, you can add 3 to both sides to get x + 3 + 3 = 7 + 3, which simplifies to x = 10.

Q: What are some common mistakes to avoid when using the Addition Property of Equality?

A: Some common mistakes to avoid when using the Addition Property of Equality include:

  • Adding different values to both sides of the equation
  • Subtracting the same value from both sides of the equation
  • Not simplifying the equation after adding or subtracting the same value

Q: How do I solve a two-step equation?

A: To solve a two-step equation, you need to follow these steps:

  1. Simplify the equation by adding or subtracting the same value from both sides.
  2. Isolate the variable by multiplying or dividing both sides of the equation by the same value.
  3. Check your solution by plugging it back into the original equation.

Q: What are some real-world applications of the Addition Property of Equality?

A: The Addition Property of Equality has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the Addition Property of Equality is used to calculate the total energy of a system, while in engineering, it is used to design and optimize systems. In economics, the Addition Property of Equality is used to calculate the total cost of a product or service.

Q: Can I use the Addition Property of Equality with fractions?

A: Yes, you can use the Addition Property of Equality with fractions. For example, if you have the equation 1/2 + x = 3/4, you can add 1/2 to both sides to get 1/2 + x + 1/2 = 3/4 + 1/2, which simplifies to x = 1/4.

Q: Can I use the Addition Property of Equality with decimals?

A: Yes, you can use the Addition Property of Equality with decimals. For example, if you have the equation 2.5 + x = 4.2, you can add 2.5 to both sides to get 2.5 + x + 2.5 = 4.2 + 2.5, which simplifies to x = 1.7.

Q: Can I use the Addition Property of Equality with negative numbers?

A: Yes, you can use the Addition Property of Equality with negative numbers. For example, if you have the equation -3 + x = 2, you can add -3 to both sides to get -3 + x - 3 = 2 - 3, which simplifies to x = 5.

Conclusion

In conclusion, the Addition Property of Equality is a fundamental concept in mathematics that helps us solve equations and inequalities. By understanding and applying this property, we can simplify equations and isolate the variable. Remember to always simplify the equation after adding or subtracting the same value, and avoid common mistakes such as adding different values to both sides of the equation.