Question

for each expression, select all equivalent expressions from the list.
(a) ( 7x + 49 )
( square 7(x + 7) )
( square 7 cdot x + 7 cdot 7 )
( square 56x )
( square 7(7x + 1) )
(b) ( 15 + 12y - 7 - y )
( square 8y + 11y )
( square 11 + 8y )
( square 8y + 11 )
( square 11y + 8 )

Explanation:

Part (a): Analyze \(7x + 49\)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \(7x\) and \(49\) is \(7\). Factor out \(7\) from the expression:
\[ 7x + 49 = 7(x + 7) \]

Step 2: Check other expressions

- \(7(x + 7)\): Matches the factored form, so it's equivalent.
- \(7\cdot x + 7\cdot 7\): By the distributive property, \(7(x + 7)=7x + 49 = 7\cdot x + 7\cdot 7\), so it's equivalent.
- \(56x\): \(7x + 49\) is not equal to \(56x\) (e.g., if \(x = 1\), \(7(1)+49 = 56\), \(56(1)=56\) – wait, no, \(7x + 49\) when \(x = 1\) is \(56\), but \(56x\) is \(56x\), which is only equal when \(x = 1\), not for all \(x\). Wait, my mistake earlier. Wait, \(7x + 49\) vs \(56x\): Let's take \(x = 2\). \(7(2)+49 = 14 + 49 = 63\), \(56(2)=112\). Not equal. So \(56x\) is not equivalent.
- \(7(7x + 1)\): Expand it: \(49x + 7\), which is not equal to \(7x + 49\) (e.g., \(x = 1\): \(49 + 7 = 56\), \(7 + 49 = 56\) – wait, \(x = 1\) gives same, but \(x = 2\): \(49(2)+7 = 105\), \(7(2)+49 = 63\). Not equal. So only \(7(x + 7)\) and \(7\cdot x + 7\cdot 7\) are equivalent to \(7x + 49\).

Part (b): Analyze \(15 + 12y - 7 - y\)

Step 1: Simplify the expression

Combine like terms:
\[ 15 - 7 + 12y - y = 8 + 11y \]

Step 2: Rewrite and check

- \(8y + 11y\): \(8y + 11y = 19y\), not equal to \(8 + 11y\).
- \(11 + 8y\): Not equal (e.g., \(y = 1\): \(11 + 8 = 19\), \(8 + 11(1)=19\) – wait, no, \(15 + 12y - 7 - y = 8 + 11y\). \(11 + 8y\) is different (coefficients of \(y\) and constants).
- \(11 + 8y\): Wait, no. Wait, \(15 + 12y - 7 - y = (15 - 7)+(12y - y)=8 + 11y = 11y + 8\). Also, \(8y + 11y = 19y\) (no), \(11 + 8y\) (no), \(11y + 8\) (yes), \(8y + 11\) (no), \(11y + 8\) is same as \(8 + 11y\), and also \(8y + 11y\) is wrong, \(8y + 11\) is wrong, \(11 + 8y\) is wrong. Wait, let's do it again:

Original expression: \(15 + 12y - 7 - y\)

Combine constants: \(15 - 7 = 8\)

Combine \(y\) terms: \(12y - y = 11y\)

So simplified: \(8 + 11y = 11y + 8\)

Now check the options:
- \(8y + 11y\): \(19y\) ≠ \(8 + 11y\)
- \(11 + 8y\): ≠ \(8 + 11y\) (constants and coefficients differ)
- \(11y + 8\): Same as \(8 + 11y\), so equivalent.
- \(8y + 11\): ≠ \(8 + 11y\)
- \(11y + 8\): Equivalent, and also \(11y + 8\) is same as \(8 + 11y\), and also \(8y + 11y\) is wrong, \(11 + 8y\) is wrong. Wait, the options are: \(8y + 11y\), \(11 + 8y\), \(11 + 8y\)? Wait, the options given are: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, the image shows for part (b) the options are: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, no, the original problem's part (b) options (from the image) are: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, the user's image has for part (b) the expressions: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, no, looking back:

Part (b) original expression: \(15 + 12y - 7 - y\)

Options: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, the image shows:

First column (part a): \(7x + 49\), then \(7(x + 7)\), \(7\cdot x + 7\cdot 7\), \(56x\), \(7(7x + 1)\)

Second column (part b): \(15 + 12y - 7 - y\), then \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? Wait, no, the second column's first row (under part b) is \(8y + 11y\)? No, the second column (part b) has:

First row (after the original expression): \(8y + 11y\)

Second row: \(11 + 8y\)

Third row: \(11y + 8\)

Fourth row: \(11y + 8\)? Wait, no, the user's image:

Part (b) original: \(15 + 12y - 7 - y\)

Then the options (second column) are:

- \(8y + 11y\)

- \(11 + 8y\)

- \(11y + 8\)

- \(11y + 8\)? Wait, no, the second column (part b) has four expressions: \(8y + 11y\), \(11 + 8y\), \(11y + 8\), \(11y + 8\)? No, probably a typo, but let's re - do:

Simplify \(15 + 12y - 7 - y\):

\(15 - 7=8\), \(12y - y = 11y\), so the simplified form is \(8 + 11y\) or \(11y + 8\).

Now check the options:

- \(8y + 11y\): \(19y eq8 + 11y\)

- \(11 + 8y\): \(11 + 8y eq8 + 11y\) (coefficients of \(y\) and constants are different)

- \(11y + 8\): This is equal to \(8 + 11y\), so it's equivalent.

- The other \(11y + 8\) (if there are two, both are equivalent). Also, \(8y + 11y\) is \(19y\), not equivalent. \(11 + 8y\) is not equivalent. So the equivalent expressions for \(15 + 12y - 7 - y\) are \(11y + 8\) (and the other \(11y + 8\) if present).

Answer:

Part (a) (for \(7x + 49\)):

• \(7(x + 7)\) (check the box)
• \(7\cdot x + 7\cdot 7\) (check the box)

Part (b) (for \(15 + 12y - 7 - y\)):

• \(11y + 8\) (check the box for both instances if present, and also note that \(8 + 11y\) is same as \(11y + 8\), so the \(11y + 8\) options are equivalent)