Question

substitute ( w = 1 ) and ( w = 3 ) to determine if the two expressions are equivalent. ( 4(3w + 4) ) ( 16w + 12 ) which statements are true? check all that apply. (square) the value of both expressions when ( w = 1 ) is 28. (square) the two expressions are equivalent. (square) the value of both expressions when ( w = 1 ) is 16. (square) the two expressions are not equivalent. (square) the value of both expressions when ( w = 3 ) is 52. (square) the value of both expressions when ( w = 3 ) is 60.

Explanation:

Step1: Simplify \(4(3w + 4)\)

Using the distributive property \(a(b + c)=ab+ac\), we get \(4\times3w+4\times4 = 12w + 16\). Wait, no, wait, the other expression is \(16w + 12\)? Wait, no, maybe I misread. Wait, the two expressions: let's check again. Wait, the first expression is \(4(3w + 4)\), the second is \(16w + 12\)? Wait, no, maybe the user's image has a typo? Wait, no, let's do substitution for \(w = 1\) and \(w = 3\).

First, for \(w = 1\):

Step2: Evaluate \(4(3w + 4)\) when \(w = 1\)

Substitute \(w = 1\): \(4(3\times1 + 4)=4(3 + 4)=4\times7 = 28\).

Step3: Evaluate \(16w + 12\) when \(w = 1\)

Substitute \(w = 1\): \(16\times1 + 12 = 16 + 12 = 28\). Wait, that's 28? But option C says 16. Wait, maybe the second expression is different? Wait, maybe the second expression is \(12w + 16\)? No, the user's image shows \(16w + 12\). Wait, maybe I made a mistake. Wait, let's check \(w = 3\):

Step4: Evaluate \(4(3w + 4)\) when \(w = 3\)

\(4(3\times3 + 4)=4(9 + 4)=4\times13 = 52\).

Step5: Evaluate \(16w + 12\) when \(w = 3\)

\(16\times3 + 12 = 48 + 12 = 60\).

Now, check the options:

• Option A: When \(w = 1\), \(4(3w + 4)=28\), \(16w + 12 = 28\). So this is true.
• Option B: Are they equivalent? Let's simplify \(4(3w + 4)=12w + 16\). The other is \(16w + 12\). \(12w + 16\) vs \(16w + 12\). These are not equivalent (since coefficients of \(w\) are different: 12 vs 16, constants 16 vs 12). So B is false.
• Option C: When \(w = 1\), \(16w + 12 = 28\), not 16. So C is false.
• Option D: Since \(12w + 16\) (from \(4(3w + 4)\)) and \(16w + 12\) are not equivalent (different linear terms), D is true.
• Option E: When \(w = 3\), \(4(3w + 4)=52\), \(16w + 12 = 60\). So E is false (52 vs 60).
• Option F: When \(w = 3\), \(16w + 12 = 60\), but \(4(3w + 4)=52\), so F is false.

Wait, but earlier when \(w = 1\), both are 28, so A is true. D is true (they are not equivalent). Wait, but let's re - check the expressions. Wait, maybe the second expression is \(12w + 16\)? No, the user's image shows \(16w + 12\). So:

For \(w = 1\):

\(4(3w + 4)=4(3 + 4)=28\)

\(16w + 12=16 + 12 = 28\) → so A is true.

For \(w = 3\):

\(4(3w + 4)=4(9 + 4)=52\)

\(16w + 12=48 + 12 = 60\)

Now, are the expressions equivalent? Let's set \(4(3w + 4)=16w + 12\)

\(12w + 16 = 16w + 12\)

\(16 - 12 = 16w - 12w\)

\(4 = 4w\) → \(w = 1\). So they are equal only when \(w = 1\), not for all \(w\), so they are not equivalent (D is true).

Now, check the options:

• A: True (both are 28 when \(w = 1\))
• B: False (not equivalent)
• C: False (both are 28 when \(w = 1\), not 16)
• D: True (not equivalent)
• E: False ( \(4(3w + 4)=52\), \(16w + 12 = 60\) when \(w = 3\), so not 52 for the second)
• F: False ( \(16w + 12 = 60\) when \(w = 3\), but \(4(3w + 4)=52\))

So the true options are A and D. Wait, but let's re - check the option text:

Option A: "The value of both expressions when \(w = 1\) is 28." → True.

Option D: "The two expressions are not equivalent." → True.

Wait, but when \(w = 1\), both are 28, when \(w = 3\), they are 52 and 60. So they are not equivalent (D is true), and A is true.

Answer:

A. The value of both expressions when \(w = 1\) is 28.
D. The two expressions are not equivalent.