Which Equation Is True?A. \[$(6-2)-7 = 6-(2-7)\$\] B. \[$(6-2)+7 = 6-(2+7)\$\] C. \[$(6 \div 2) \div 7 = 6 \div (2 \div 7)\$\] D. \[$(6+2)+7 = 6+(2+7)\$\]

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In mathematics, equations are used to represent relationships between variables. When evaluating equations, it's essential to follow the order of operations (PEMDAS) to ensure accuracy. In this article, we'll examine four equations and determine which one is true.

Equation A: (6−2)−7=6−(2−7)(6-2)-7 = 6-(2-7)

Let's start by evaluating Equation A: (6−2)−7=6−(2−7)(6-2)-7 = 6-(2-7). To do this, we need to follow the order of operations.

First, we'll evaluate the expressions inside the parentheses:

  • (6−2)=4(6-2) = 4
  • (2−7)=−5(2-7) = -5

Now, we can rewrite the equation as:

  • 4−7=6−(−5)4-7 = 6-(-5)

Next, we'll evaluate the expressions on the right-hand side:

  • 6−(−5)=6+5=116-(-5) = 6+5 = 11

So, the equation becomes:

  • 4−7=114-7 = 11

However, this is not true. The correct result is:

  • 4−7=−34-7 = -3

Therefore, Equation A is false.

Equation B: (6−2)+7=6−(2+7)(6-2)+7 = 6-(2+7)

Now, let's evaluate Equation B: (6−2)+7=6−(2+7)(6-2)+7 = 6-(2+7). Again, we'll follow the order of operations.

First, we'll evaluate the expressions inside the parentheses:

  • (6−2)=4(6-2) = 4
  • (2+7)=9(2+7) = 9

Now, we can rewrite the equation as:

  • 4+7=6−94+7 = 6-9

Next, we'll evaluate the expressions on the right-hand side:

  • 6−9=−36-9 = -3

So, the equation becomes:

  • 4+7=−34+7 = -3

However, this is not true. The correct result is:

  • 4+7=114+7 = 11

Therefore, Equation B is false.

Equation C: (6÷2)÷7=6÷(2÷7)(6 \div 2) \div 7 = 6 \div (2 \div 7)

Now, let's evaluate Equation C: (6÷2)÷7=6÷(2÷7)(6 \div 2) \div 7 = 6 \div (2 \div 7). Again, we'll follow the order of operations.

First, we'll evaluate the expressions inside the parentheses:

  • (6÷2)=3(6 \div 2) = 3
  • (2÷7)=2/7(2 \div 7) = 2/7

Now, we can rewrite the equation as:

  • 3÷7=6÷(2/7)3 \div 7 = 6 \div (2/7)

Next, we'll evaluate the expressions on the right-hand side:

  • 6÷(2/7)=6×(7/2)=216 \div (2/7) = 6 \times (7/2) = 21

So, the equation becomes:

  • 3÷7=213 \div 7 = 21

However, this is not true. The correct result is:

  • 3÷7=3/73 \div 7 = 3/7

Therefore, Equation C is false.

Equation D: (6+2)+7=6+(2+7)(6+2)+7 = 6+(2+7)

Finally, let's evaluate Equation D: (6+2)+7=6+(2+7)(6+2)+7 = 6+(2+7). Again, we'll follow the order of operations.

First, we'll evaluate the expressions inside the parentheses:

  • (6+2)=8(6+2) = 8
  • (2+7)=9(2+7) = 9

Now, we can rewrite the equation as:

  • 8+7=6+98+7 = 6+9

Next, we'll evaluate the expressions on the right-hand side:

  • 6+9=156+9 = 15

So, the equation becomes:

  • 8+7=158+7 = 15

This is true. The correct result is indeed:

  • 8+7=158+7 = 15

Therefore, Equation D is true.

Conclusion

In conclusion, we've evaluated four equations and determined which one is true. Equation D: (6+2)+7=6+(2+7)(6+2)+7 = 6+(2+7) is the correct equation. It's essential to follow the order of operations when evaluating equations to ensure accuracy.

Order of Operations (PEMDAS)

When evaluating equations, it's crucial to follow the order of operations (PEMDAS):

  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, you can ensure that your equations are evaluated accurately and consistently.

Common Mistakes

When evaluating equations, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
  • Not evaluating expressions inside parentheses first: Failing to evaluate expressions inside parentheses first can lead to incorrect results.
  • Not evaluating exponential expressions next: Failing to evaluate exponential expressions next can lead to incorrect results.

By avoiding these common mistakes, you can ensure that your equations are evaluated accurately and consistently.

Real-World Applications

Equations are used in a wide range of real-world applications, including:

  • Science: Equations are used to model and predict the behavior of physical systems.
  • Engineering: Equations are used to design and optimize systems.
  • Finance: Equations are used to model and predict financial systems.
  • Computer Science: Equations are used to model and predict the behavior of algorithms.

By understanding how to evaluate equations, you can apply this knowledge to a wide range of real-world applications.

Final Thoughts

In the previous article, we evaluated four equations and determined which one is true. In this article, we'll answer some frequently asked questions about equation evaluation.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

  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.

Q: Why is it important to follow the order of operations?

A: Following the order of operations is crucial to ensure that mathematical expressions are evaluated accurately and consistently. If the order of operations is not followed, it can lead to incorrect results.

Q: What happens if I don't follow the order of operations?

A: If you don't follow the order of operations, you may get incorrect results. For example, if you evaluate the expression 3 + 4 × 5 without following the order of operations, you may get 7 × 5 = 35, which is incorrect. The correct result is 3 + 20 = 23.

Q: How do I evaluate expressions with multiple operations?

A: To evaluate expressions with multiple operations, follow the order of operations:

  1. Evaluate any expressions inside parentheses first.
  2. Evaluate any exponential expressions next.
  3. Evaluate any multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between multiplication and division?

A: Multiplication and division are both arithmetic operations that involve numbers. However, multiplication involves combining numbers to get a product, while division involves splitting a number into equal parts.

Q: How do I evaluate expressions with fractions?

A: To evaluate expressions with fractions, follow the order of operations:

  1. Evaluate any expressions inside parentheses first.
  2. Evaluate any exponential expressions next.
  3. Evaluate any multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a number as a sum of powers of 10.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, divide the numerator by the denominator.

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

A: An equation is a statement that two expressions are equal, while an expression is a mathematical statement that contains variables and constants.

Q: How do I solve an equation?

A: To solve an equation, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding, subtracting, 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 is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding, subtracting, 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: How do I solve a quadratic equation?

A: To solve a quadratic equation, follow these steps:

  1. Simplify the equation by combining like terms.
  2. Factor the equation, if possible.
  3. Use the quadratic formula to find the solutions.
  4. Check your solutions by plugging them back into the original equation.

Conclusion

In conclusion, equation evaluation is a crucial skill in mathematics. By following the order of operations and understanding how to evaluate expressions with multiple operations, fractions, and decimals, you can ensure that your equations are evaluated accurately and consistently. Whether you're working in science, engineering, finance, or computer science, understanding how to evaluate equations is essential for success.