Select All Equations That Are Equivalent To $0.6 + 15b + 4 = 25.6$.A. $15b + 4 = 25.6$B. $15b + 4 = 25$C. $3(0.6 + 15b + 4) = 76.8$D. $15b = 25.6$E. $15b = 21$

Understanding Equivalent Equations

Equivalent equations are mathematical expressions that have the same solution or value. In other words, if two equations are equivalent, they will have the same set of solutions or values for the variables involved. In this article, we will focus on selecting equivalent equations to the given equation $0.6 + 15b + 4 = 25.6$.

The Given Equation

The given equation is $0.6 + 15b + 4 = 25.6$. This equation involves a constant term, a variable term ($15b$), and an equal sign. To find equivalent equations, we need to manipulate this equation using basic algebraic operations.

Step 1: Simplify the Equation

The first step in finding equivalent equations is to simplify the given equation. We can start by combining the constant terms on the left-hand side of the equation.

0.6 + 4 = 4.6

Now, the equation becomes $4.6 + 15b = 25.6$. This simplification is a basic algebraic operation that helps us to make the equation more manageable.

Step 2: Isolate the Variable Term

The next step is to isolate the variable term ($15b$) on one side of the equation. We can do this by subtracting $4.6$ from both sides of the equation.

4.6 + 15b = 25.6
15b = 25.6 - 4.6
15b = 21

Now, the equation becomes $15b = 21$. This is a simplified version of the original equation.

Step 3: Identify Equivalent Equations

Now that we have simplified the equation, we can identify equivalent equations. Equivalent equations have the same solution or value as the original equation. Let's examine the answer choices and determine which ones are equivalent to the simplified equation $15b = 21$.

A. $15b + 4 = 25.6$

This equation is not equivalent to the simplified equation $15b = 21$. The variable term ($15b$) is present on both sides of the equation, but the constant term ($4$) is not isolated on one side.

B. $15b + 4 = 25$

This equation is not equivalent to the simplified equation $15b = 21$. The constant term ($4$) is not isolated on one side, and the right-hand side of the equation is different from the simplified equation.

C. $3(0.6 + 15b + 4) = 76.8$

This equation is not equivalent to the simplified equation $15b = 21$. The equation involves a multiplication operation and a different constant term.

D. $15b = 25.6$

This equation is not equivalent to the simplified equation $15b = 21$. The right-hand side of the equation is different from the simplified equation.

E. $15b = 21$

This equation is equivalent to the simplified equation $15b = 21$. The variable term ($15b$) is isolated on one side, and the constant term is the same as the simplified equation.

Conclusion

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Q: What is an equivalent equation?

A: An equivalent equation is a mathematical expression that has the same solution or value as the original equation. In other words, if two equations are equivalent, they will have the same set of solutions or values for the variables involved.

Q: How do I determine if two equations are equivalent?

A: To determine if two equations are equivalent, you need to check if they have the same solution or value. You can do this by simplifying both equations and comparing the resulting expressions.

Q: What are some common ways to manipulate equations to find equivalent equations?

A: Some common ways to manipulate equations to find equivalent equations include:

• Adding or subtracting the same value to both sides of the equation
• Multiplying or dividing both sides of the equation by the same non-zero value
• Combining like terms on the same side of the equation
• Isolating the variable term on one side of the equation

Q: Can you give an example of how to find equivalent equations using these methods?

A: Let's consider the equation $2x + 3 = 7$. To find equivalent equations, we can start by isolating the variable term ($2x$) on one side of the equation.

2x + 3 = 7
2x = 7 - 3
2x = 4

Now, we have the equation $2x = 4$. This is a simplified version of the original equation. We can find equivalent equations by manipulating this equation using the methods mentioned earlier.

Q: How do I know if an equation is equivalent to the original equation?

A: To determine if an equation is equivalent to the original equation, you need to check if it has the same solution or value. You can do this by simplifying both equations and comparing the resulting expressions.

Q: What are some common mistakes to avoid when finding equivalent equations?

A: Some common mistakes to avoid when finding equivalent equations include:

• Not simplifying the equation enough to reveal the equivalent equation
• Not checking if the equivalent equation has the same solution or value as the original equation
• Not considering all possible equivalent equations

Q: Can you give an example of a common mistake to avoid when finding equivalent equations?

A: Let's consider the equation $2x + 3 = 7$. If we simplify the equation by isolating the variable term ($2x$) on one side, we get $2x = 4$. However, if we don't simplify the equation enough, we might get an equation like $2x + 1 = 5$, which is not equivalent to the original equation.

Q: How do I know if an equation is not equivalent to the original equation?

A: To determine if an equation is not equivalent to the original equation, you need to check if it has a different solution or value. You can do this by simplifying both equations and comparing the resulting expressions.

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

A: Equivalent equations have many real-world applications, including:

• Solving systems of linear equations
• Finding the equation of a line
• Solving quadratic equations
• Modeling real-world phenomena using mathematical equations

Q: Can you give an example of a real-world application of equivalent equations?

A: Let's consider a scenario where a company wants to determine the cost of producing a certain number of units of a product. The company has a linear equation that represents the cost of production, and they want to find the equivalent equation that represents the cost of producing a certain number of units. By manipulating the equation using equivalent equations, the company can find the equivalent equation that represents the cost of producing a certain number of units.