4. In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms. (a) , , , (b) , , , , ,

Understand the Question An arithmetic expression is made up of terms, where each term has a specific sign ( or ). By the commutative property of addition, rearranging the terms along with their signs does not change the total value (e.g., ). When expanding brackets preceded by a minus sign, the sign of each term inside changes: . Without performing calculations, expressions have the same value if they consist of the exact same terms with the same signs. Part (a) , , , Step 1 · Compare Terms and Signs Identify the terms and their signs in each expression: : Terms are and : Terms are and : Terms are and : Terms are and By the commutative property of addition, and have the identical terms and . Answer > (a) and Part (b) , , , , , Step 1 · Compare Terms and Signs Identify the terms and expand brackets in each expression: : Terms are , , and (repeated three times) : Terms are , , and : Terms are , , and : Terms are , , and Grouping expressions with identical terms: 1. The three identical instances of have the same value. 2. and both have terms , , and , so they have the same value. Answer > (b) , , and have the same value. Also, and have the same value. Common Mistakes Order in Subtraction: Assuming . Subtraction is not commutative, so . Sign Error with Parentheses: Incorrectly expanding as instead of .

Arithmetic Expressions

Question 4

In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms.

(a) $319 + 537$ , $319 - 537$ , $-537 + 319$ , $537 - 319$

(b) $87 + 46 - 109$ , $87 + 46 - 109$ , $87 + 46 - 109$ , $87 - 46 + 109$ , $87 - (46 + 109)$ , $(87 - 46) + 109$

Solution

Understand the Question

An arithmetic expression is made up of terms, where each term has a specific sign ( $+$ or $-$ ).
By the commutative property of addition, rearranging the terms along with their signs does not change the total value (e.g., $a - b = -b + a$ ).
When expanding brackets preceded by a minus sign, the sign of each term inside changes: $a - (b + c) = a - b - c$ .
Without performing calculations, expressions have the same value if they consist of the exact same terms with the same signs.

(a) $319 + 537$ , $319 - 537$ , $-537 + 319$ , $537 - 319$

Step 1 · Compare Terms and Signs

Diagram 1

Identify the terms and their signs in each expression:

$319 + 537$ : Terms are $+319$ and $+537$
$319 - 537 = 319 + (-537)$ : Terms are $+319$ and $-537$
$-537 + 319 = 319 + (-537)$ : Terms are $+319$ and $-537$
$537 - 319 = 537 + (-319)$ : Terms are $+537$ and $-319$

By the commutative property of addition, $319 - 537$ and $-537 + 319$ have the identical terms $+319$ and $-537$ .

Answer

(a) $319 - 537$ and $-537 + 319$

(b) $87 + 46 - 109$ , $87 + 46 - 109$ , $87 + 46 - 109$ , $87 - 46 + 109$ , $87 - (46 + 109)$ , $(87 - 46) + 109$

Step 1 · Compare Terms and Signs

Identify the terms and expand brackets in each expression:

$87 + 46 - 109$ : Terms are $+87$ , $+46$ , and $-109$ (repeated three times)
$87 - 46 + 109$ : Terms are $+87$ , $-46$ , and $+109$
$87 - (46 + 109) = 87 - 46 - 109$ : Terms are $+87$ , $-46$ , and $-109$
$(87 - 46) + 109 = 87 - 46 + 109$ : Terms are $+87$ , $-46$ , and $+109$

Grouping expressions with identical terms:

The three identical instances of $87 + 46 - 109$ have the same value.
$87 - 46 + 109$ and $(87 - 46) + 109$ both have terms $+87$ , $-46$ , and $+109$ , so they have the same value.

Answer

(b) $87 + 46 - 109$ , $87 + 46 - 109$ , and $87 + 46 - 109$ have the same value. Also, $87 - 46 + 109$ and $(87 - 46) + 109$ have the same value.

Common Mistakes

Order in Subtraction: Assuming $a - b = b - a$ . Subtraction is not commutative, so $319 - 537 \neq 537 - 319$ .
Sign Error with Parentheses: Incorrectly expanding $87 - (46 + 109)$ as $87 - 46 + 109$ instead of $87 - 46 - 109$ .

Question 4

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