Ask a New Question
Menu

Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.

Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11

3 answers

To demonstrate that Expression 1 is equivalent to Expression 2 using the Associative Property, we first rewrite both expressions clearly:

Expression 1: \( 22r - 13 \)

Expression 2: \( 19r + (3r - 2) - 11 \)

We will simplify Expression 2 step by step:

  1. Start with Expression 2: \[ 19r + (3r - 2) - 11 \]

  2. Distribute and simplify within the parentheses: \[ = 19r + 3r - 2 - 11 \]

  3. Combine like terms: \[ = (19r + 3r) + (-2 - 11) \] \[ = 22r - 13 \]

Now, we see that Expression 2 simplifies to \( 22r - 13 \), which is exactly the same as Expression 1.

Thus, using the Associative Property (which states that the way in which numbers are grouped does not change their sum or product), we have shown that:

Expression 1 \( 22r - 13 \) is equivalent to Expression 2 \( 19r + (3r - 2) - 11 \).

Therefore, both expressions are equivalent.

can you explain why ?

Certainly! Let's walk through the reasoning step-by-step to understand why Expression 1 and Expression 2 are equivalent.

Expressions

  1. Expression 1: \[ 22r - 13 \]

  2. Expression 2: \[ 19r + (3r - 2) - 11 \]

Step-by-Step Breakdown

We need to simplify Expression 2 and show that it equals Expression 1.

Step 1: Distributing and Combining the Terms

  • Start with Expression 2: \[ 19r + (3r - 2) - 11 \]

  • The parentheses around \( (3r - 2) \) can be removed directly since there's nothing multiplying the parentheses that would change the terms inside:

    Simplifying gives: \[ 19r + 3r - 2 - 11 \]

Step 2: Combine Like Terms

  • Now, we will group and combine the like terms. The terms involving \( r \) are \( 19r \) and \( 3r \), and the constant terms are \(-2\) and \(-11\).

  • Combine the \( r \) terms: \[ 19r + 3r = 22r \]

  • Combine the constant terms: \[ -2 - 11 = -13 \]

Step 3: Write the Simplified Expression

Now we can write the total simplified expression: \[ 22r - 13 \]

Conclusion

We have simplified Expression 2 down to: \[ 22r - 13 \] which is exactly the same as Expression 1.

Why are they Equivalent?

  1. Mathematical Equivalence: Two expressions are mathematically equivalent if they yield the same value for the same inputs. Here, both expressions produce the same result for any value of \( r \).

  2. Associative Property: The associative property primarily deals with the grouping of numbers in addition and multiplication. While in this case, we didn't explicitly group numbers, the property assures us that changing the order or grouping of addition and subtraction doesn't change the result.

  3. Combination of Like Terms: By rearranging terms (which the associative property also supports), we are allowed to combine like terms directly, which leads us back to the same final expression.

Thus, both expressions represent the same mathematical relationship, showing that \( 22r - 13 \) is indeed equivalent to \( 19r + (3r - 2) - 11 \). This aligns with fundamental properties of arithmetic operations, confirming their equality regardless of how they are expressed.