Here Is An Equation That Is True For All Values Of $x$: $5(x+2)=5x+10$. Elena Saw This Equation And Says She Can Tell $20(x+2)+31=4(5x+10)+31$ Is Also True For Any Value Of $x$. How Can She Tell? Explain Your Reasoning.

Elena's Insight: A Deeper Look into Algebraic Equations

In the world of mathematics, equations are the building blocks of problem-solving. They allow us to express relationships between variables and constants, and to solve for unknown values. Elena, a math enthusiast, has stumbled upon an equation that she believes is true for any value of $x$. The equation is $20(x+2)+31=4(5x+10)+31$. But how can she be so sure? In this article, we will delve into the world of algebra and explore the reasoning behind Elena's insight.

Let's start with the original equation that Elena saw: $5(x+2)=5x+10$. This equation is true for all values of $x$. To see why, let's expand the left-hand side of the equation:

$$5(x+2) = 5x + 10$$

Using the distributive property, we can rewrite the left-hand side as:

$$5x + 10 = 5x + 10$$

As we can see, the left-hand side is equal to the right-hand side, and this equation is true for all values of $x$.

Now, let's take a look at the equation that Elena believes is true: $20(x+2)+31=4(5x+10)+31$. At first glance, this equation may seem daunting, but let's break it down step by step.

Step 1: Expand the Left-Hand Side

Using the distributive property, we can expand the left-hand side of the equation:

$$20(x+2) + 31 = 20x + 40 + 31$$

Simplifying the expression, we get:

$$20x + 71$$

Step 2: Expand the Right-Hand Side

Using the distributive property, we can expand the right-hand side of the equation:

$$4(5x+10) + 31 = 20x + 40 + 31$$

Simplifying the expression, we get:

$$20x + 71$$

Step 3: Compare the Left-Hand Side and Right-Hand Side

Now that we have expanded both sides of the equation, we can compare them. As we can see, the left-hand side and right-hand side are equal:

$$20x + 71 = 20x + 71$$

This equation is true for all values of $x$, just like the original equation that Elena saw.

Elena's insight into the equation $20(x+2)+31=4(5x+10)+31$ is based on her understanding of algebraic equations. By expanding both sides of the equation and comparing them, she can see that the equation is true for all values of $x$. This is a great example of how algebra can be used to solve problems and understand relationships between variables and constants.

Algebra is a powerful tool that can be used to solve a wide range of problems. By understanding the rules of algebra, such as the distributive property and the commutative property, we can solve equations and inequalities with ease. In this article, we have seen how Elena used her knowledge of algebra to understand the equation $20(x+2)+31=4(5x+10)+31$. This is just one example of how algebra can be used to solve problems and understand relationships between variables and constants.

Algebra has many real-world applications. For example, it can be used to solve problems in physics, engineering, and economics. In physics, algebra can be used to describe the motion of objects and to solve problems involving forces and energies. In engineering, algebra can be used to design and optimize systems, such as bridges and buildings. In economics, algebra can be used to model and analyze economic systems, such as supply and demand.

In conclusion, Elena's insight into the equation $20(x+2)+31=4(5x+10)+31$ is based on her understanding of algebraic equations. By expanding both sides of the equation and comparing them, she can see that the equation is true for all values of $x$. This is a great example of how algebra can be used to solve problems and understand relationships between variables and constants. Whether you are a math enthusiast like Elena or just starting to learn algebra, this article has shown you the power of algebra and its many real-world applications.

• [1] "Algebra" by Michael Artin
• [2] "Introduction to Algebra" by Richard Rusczyk
• [3] "Algebra and Trigonometry" by James Stewart

• Algebra: A branch of mathematics that deals with the study of variables and their relationships.
• Equation: A statement that two expressions are equal.
• Variable: A symbol that represents a value that can change.
• Constant: A value that does not change.
• Distributive property: A property of algebra that states that a single value can be distributed to multiple values.
• Commutative property: A property of algebra that states that the order of values does not change the result.
  Elena's Insight: A Deeper Look into Algebraic Equations - Q&A

In our previous article, we explored the equation $20(x+2)+31=4(5x+10)+31$ and how Elena, a math enthusiast, was able to determine that it is true for any value of $x$. In this article, we will answer some of the most frequently asked questions about this equation and provide additional insights into the world of algebra.

Q: What is the distributive property, and how is it used in this equation?

A: The distributive property is a fundamental concept in algebra that states that a single value can be distributed to multiple values. In the equation $20(x+2)+31=4(5x+10)+31$, the distributive property is used to expand the left-hand side of the equation:

$$20(x+2) = 20x + 40$$

This allows us to simplify the equation and compare it to the right-hand side.

Q: How can I apply the distributive property to other equations?

A: The distributive property can be applied to any equation that involves the multiplication of a single value by multiple values. For example, consider the equation $3(2x+5)$. Using the distributive property, we can expand this equation as follows:

$$3(2x+5) = 6x + 15$$

This allows us to simplify the equation and solve for the value of $x$.

Q: What is the commutative property, and how is it used in this equation?

A: The commutative property is a fundamental concept in algebra that states that the order of values does not change the result. In the equation $20(x+2)+31=4(5x+10)+31$, the commutative property is not explicitly used, but it is an underlying principle that allows us to simplify the equation and compare it to the right-hand side.

Q: How can I apply the commutative property to other equations?

A: The commutative property can be applied to any equation that involves the addition or multiplication of values. For example, consider the equation $2x + 5 = 5 + 2x$. Using the commutative property, we can rewrite this equation as follows:

$$2x + 5 = 5 + 2x$$

This allows us to simplify the equation and solve for the value of $x$.

Q: What is the difference between an equation and an inequality?

A: An equation is a statement that two expressions are equal, while an inequality is a statement that one expression is greater than or less than another expression. In the equation $20(x+2)+31=4(5x+10)+31$, we are dealing with an equation, while in an inequality, we would be dealing with a statement that one expression is greater than or less than another expression.

Q: How can I solve an inequality?

A: To solve an inequality, we can use the same techniques that we use to solve an equation. However, we must also consider the direction of the inequality. For example, consider the inequality $2x + 5 > 5 + 2x$. To solve this inequality, we can subtract $2x$ from both sides and simplify the equation:

$$5 > 5$$

This tells us that the inequality is true for all values of $x$.

Q: What is the significance of the constant term in an equation?

A: The constant term in an equation is a value that does not change. In the equation $20(x+2)+31=4(5x+10)+31$, the constant term is $31$. This value does not change, regardless of the value of $x$.

Q: How can I use the constant term to simplify an equation?

A: The constant term can be used to simplify an equation by combining like terms. For example, consider the equation $2x + 5 = 5 + 2x$. We can combine the constant terms on the left-hand side and right-hand side of the equation:

$$2x + 5 = 5 + 2x$$

This allows us to simplify the equation and solve for the value of $x$.

In this article, we have answered some of the most frequently asked questions about the equation $20(x+2)+31=4(5x+10)+31$ and provided additional insights into the world of algebra. We have discussed the distributive property, the commutative property, and the significance of the constant term in an equation. We have also provided examples of how to apply these concepts to other equations and inequalities. Whether you are a math enthusiast like Elena or just starting to learn algebra, this article has shown you the power of algebra and its many real-world applications.

• [1] "Algebra" by Michael Artin
• [2] "Introduction to Algebra" by Richard Rusczyk
• [3] "Algebra and Trigonometry" by James Stewart

• Algebra: A branch of mathematics that deals with the study of variables and their relationships.
• Equation: A statement that two expressions are equal.
• Variable: A symbol that represents a value that can change.
• Constant: A value that does not change.
• Distributive property: A property of algebra that states that a single value can be distributed to multiple values.
• Commutative property: A property of algebra that states that the order of values does not change the result.