Question

real numbers
identifying equivalent algebraic expressions
for each expression, select all equivalent expressions from the list.
(a) $5x + 35$
$\square$ $5 \cdot x + 5 \cdot 7$
$\square$ $5(x + 7)$
$\square$ $40x$
$\square$ $5(7x + 1)$
(b) $10 + 9y - 8 - y$
$\square$ $2y + 8$
$\square$ $8 + 2y$
$\square$ $2y + 8y$
$\square$ $8y + 2$

Explanation:

Part (a): Analyze \(5x + 35\)

Step 1: Factor out common term

We can factor out 5 from \(5x + 35\). Using the distributive property \(ab+ac=a(b + c)\), here \(a = 5\), \(b=x\), \(c = 7\). So \(5x+35=5(x + 7)\).

Step 2: Expand \(5(x + 7)\)

Using the distributive property \(a(b + c)=ab+ac\), where \(a = 5\), \(b=x\), \(c = 7\), we get \(5\times x+5\times7=5x + 35\), which is the original expression. Also, \(5\cdot x+5\cdot7\) is the expanded form of \(5(x + 7)\), so \(5\cdot x+5\cdot7\) is equivalent to \(5x + 35\).

Step 3: Check other options

• For \(40x\): \(5x+35\) cannot be simplified to \(40x\) as there is a constant term 35 and the coefficient of \(x\) is 5, not 40.
• For \(5(7x + 1)\): Expanding \(5(7x+1)=35x + 5\), which is not equal to \(5x + 35\).

Part (b): Analyze \(10+9y-8 - y\)

Step 1: Combine like terms

First, combine the constant terms: \(10-8=2\). Then combine the \(y\) - terms: \(9y-y = 8y\). So \(10 + 9y-8-y=(10 - 8)+(9y-y)=2 + 8y\) or \(8y+2\) (addition is commutative). Also, \(2y + 8\) is not equivalent. Let's re - check:

• \(10+9y-8 - y=(10 - 8)+(9y - y)=2+8y\).
• \(8 + 2y\): The coefficients of \(y\) and the constant terms are different from \(2 + 8y\) (the coefficient of \(y\) in \(8 + 2y\) is 2 and in \(2+8y\) is 8 for the \(y\) - term, and constants are 8 and 2 respectively).
• \(2y+8y=10y\), which is not equal to \(2 + 8y\) (one is a pure \(y\) - term and the other has a constant term).
• \(8y + 2\) is equivalent to \(2+8y\) (by commutative property of addition).

Answer:

(a) Equivalent expressions to \(5x + 35\) are \(5(x + 7)\) and \(5\cdot x+5\cdot7\)
(b) Equivalent expressions to \(10 + 9y-8 - y\) are \(2y + 8y\) (wait, no, earlier mistake: \(10+9y - 8-y=(10 - 8)+(9y - y)=2 + 8y\), so equivalent expressions are \(2 + 8y\) (or \(8y + 2\)) and \(2y+8y\) is wrong. Correcting:
After combining like terms \(10+9y - 8-y=(10 - 8)+(9y - y)=2+8y\). So the equivalent expressions are \(2 + 8y\) (or \(8y + 2\)) and also \(2y+8y\) is incorrect. Wait, \(9y-y = 8y\) and \(10 - 8=2\), so \(10+9y-8 - y=2 + 8y\). So the equivalent expressions are \(2+8y\) (or \(8y + 2\)) and also when we rewrite \(2 + 8y\) as \(8y+2\), and also \(2y + 8\) is wrong. Wait, let's do it again:

For part (b):

\(10+9y-8 - y=(10 - 8)+(9y - y)=2+8y\)

• \(2y+8\): \(2y + 8\) has \(y\) coefficient 2 and constant 8, while \(2 + 8y\) has \(y\) coefficient 8 and constant 2. Not equivalent.
• \(2y + 8y=10y\), not equivalent.
• \(8y+2\): Equivalent to \(2 + 8y\) (commutative property of addition).
• \(8 + 2y\): Not equivalent.

So for (a): \(5(x + 7)\), \(5\cdot x+5\cdot7\)

For (b): \(8y + 2\), \(2+8y\) (which is the same as \(8y + 2\)) and also when we re - evaluate \(10+9y-8 - y=2 + 8y\), so the equivalent expressions are \(8y + 2\) and \(2+8y\) (and also \(2y+8y\) is wrong, my earlier mistake. The correct combination gives \(2 + 8y\), so the equivalent expressions are \(8y+2\) and \(2 + 8y\) (same thing) and also \(2y + 8\) is wrong. The correct equivalent expressions for (b) are \(2+8y\) (or \(8y + 2\)) and also \(2y+8y\) is incorrect. The correct way:

\(10+9y-8 - y=(10 - 8)+(9y - y)=2+8y\)

So the equivalent expressions are \(2 + 8y\) (or \(8y + 2\)) and also \(8y+2\) is the same as \(2 + 8y\). Also, \(2y+8\) is not equivalent, \(2y + 8y = 10y\) is not equivalent.

So final answer:

(a) The equivalent expressions to \(5x + 35\) are \(5(x + 7)\) and \(5\cdot x+5\cdot7\)

(b) The equivalent expressions to \(10 + 9y-8 - y\) are \(8y + 2\) (or \(2+8y\))