Question

7
which of these expressions are equivalent to $8(t - 4)$?
drag each expression to the correct category.
equivalent to $8(t - 4)$ not equivalent to $8(t - 4)$
$2(16 - 4t)$ $4(2t - 8)$ $8(4) - 8(t)$ $8(t) - 8(4)$
$2(4t - 2)$ $8t - 32$

Explanation:

First, we expand \(8(t - 4)\) using the distributive property \(a(b - c)=ab - ac\). So, \(8(t - 4)=8t - 32\). Also, by the distributive property, \(8(t - 4)=8(t)-8(4)\). Now we analyze each expression:

Step 1: Analyze \(2(16 - 4t)\)

Expand \(2(16 - 4t)\) using the distributive property \(a(b - c)=ab - ac\). So, \(2(16 - 4t)=2\times16 - 2\times4t = 32 - 8t=-8t + 32\). Now, compare with \(8t - 32\). We can factor out \(- 1\) from \(-8t + 32\) to get \(-(8t - 32)\), which is equivalent to \(-8(t - 4)\), not the same as \(8(t - 4)\) in sign, but wait, let's check again. Wait, \(8(t - 4)=8t - 32\), and \(2(16 - 4t)=32 - 8t=-(8t - 32)=-8(t - 4)\). But wait, maybe I made a mistake. Wait, let's check \(4(2t - 8)\):

Step 2: Analyze \(4(2t - 8)\)

Expand \(4(2t - 8)\) using the distributive property \(a(b - c)=ab - ac\). So, \(4(2t - 8)=4\times2t - 4\times8 = 8t - 32\), which is equal to \(8(t - 4)\).

Step 3: Analyze \(8(4)-8(t)\)

\(8(4)-8(t)=32 - 8t=-8t + 32=-(8t - 32)=-8(t - 4)\), which is not equivalent to \(8(t - 4)\) (since it has a negative sign in front).

Step 4: Analyze \(8(t)-8(4)\)

By the distributive property \(a(b - c)=ab - ac\), \(8(t)-8(4)=8(t - 4)\), so this is equivalent.

Step 5: Analyze \(2(4t - 2)\)

Expand \(2(4t - 2)\) using the distributive property \(a(b - c)=ab - ac\). So, \(2(4t - 2)=2\times4t - 2\times2 = 8t - 4\), which is not equal to \(8t - 32\).

Step 6: Analyze \(8t - 32\)

This is the expanded form of \(8(t - 4)\), so it is equivalent.

Equivalent to \(8(t - 4)\):

• \(4(2t - 8)\) (since \(4(2t - 8)=8t - 32\))
• \(8(t)-8(4)\) (by distributive property)
• \(8t - 32\) (expanded form)

Not equivalent to \(8(t - 4)\):

• \(2(16 - 4t)\) (expands to \(32 - 8t=-8t + 32\), not equal to \(8t - 32\))
• \(8(4)-8(t)\) (expands to \(32 - 8t=-8t + 32\), not equal to \(8t - 32\))
• \(2(4t - 2)\) (expands to \(8t - 4\), not equal to \(8t - 32\))

Answer:

Equivalent to \(8(t - 4)\):

• \(4(2t - 8)\)
• \(8(t) - 8(4)\)
• \(8t - 32\)

Not equivalent to \(8(t - 4)\):

• \(2(16 - 4t)\)
• \(8(4) - 8(t)\)
• \(2(4t - 2)\)