Question

real numbers
identifying equivalent algebraic expressions
for each expression, select all equivalent expressions from the list.
(a) ( x + x + x + x + x ) ( square , x + 5 ) ( square , 5x ) ( square , 5 + x ) ( square , 3x - 2x )
(b) ( 32y - 20 ) ( square , 4 cdot 8y - 4 cdot 5 ) ( square , 16y - 10 ) ( square , 12y ) ( square , 4(8y - 5) )

Explanation:

Part (a)

Step 1: Simplify \( x + x + x + x + x \)

We have 5 terms of \( x \) added together. By the definition of like terms, adding \( x \) five times is the same as multiplying \( x \) by 5. So \( x + x + x + x + x=5x \).

Step 2: Analyze other options

• \( x + 5 \): This is \( x \) plus 5, not related to 5 \( x \).
• \( 5 + x \): This is 5 plus \( x \), same as \( x + 5 \), not equivalent.
• \( 3x-2x=(3 - 2)x=x\), which is not equal to \( 5x \).

Part (b)

Step 1: Analyze \( 4\cdot8y-4\cdot5 \)

Using the distributive property \( a(b - c)=ab - ac \), here \( 4\cdot8y-4\cdot5 = 4(8y - 5)\) (by factoring out 4 from both terms). Also, \( 4\times8y=32y \) and \( 4\times5 = 20 \), so \( 4\cdot8y-4\cdot5=32y - 20 \), which is equivalent.

Step 2: Analyze \( 16y-10 \)

If we factor out 2 from \( 32y-20 \), we get \( 2(16y - 10) \), which is not equal to \( 32y-20 \) (unless \( 16y-10=32y - 20 \) which implies \( 10 = 16y \), not an identity).

Step 3: Analyze \( 12y \)

\( 32y-20 \) is a linear expression with two terms, \( 12y \) is a single term and \( 32y-20
eq12y \) (for example, if \( y = 1 \), \( 32(1)-20 = 12 \), but \( 12y=12(1) = 12 \), wait no, wait \( 32y-20=12y\) implies \( 20y=20\) or \( y = 1 \), it's not an identity. Wait, no, my mistake. Let's recalculate. \( 32y-20\) and \( 12y \): if \( y = 2 \), \( 32(2)-20=64 - 20 = 44 \), \( 12(2)=24\), so not equal.

Step 4: Analyze \( 4(8y - 5) \)

Using the distributive property, \( 4(8y-5)=4\times8y-4\times5=32y - 20 \), which is equivalent.

Answer:

(a) The equivalent expression is \( 5x \) (so the checkbox next to \( 5x \) should be selected).
(b) The equivalent expressions are \( 4\cdot8y - 4\cdot5 \) and \( 4(8y - 5) \) (so the checkboxes next to \( 4\cdot8y - 4\cdot5 \) and \( 4(8y - 5) \) should be selected).