Question

thuy is substituting t = 3 and t = 8 to determine if the two expressions are equivalent. \\(4(6t + 1)\\) \\(24t + 1\\) which statement is true? \\(\bigcirc\\) both expressions are equivalent to 73 when t = 3. \\(\bigcirc\\) both expressions are equivalent to 76 when t = 3. \\(\bigcirc\\) both expressions are equivalent to 193 when t = 8. \\(\bigcirc\\) the expressions are not equivalent.

Explanation:

Step1: Simplify \(4(6t + 1)\)

Using the distributive property \(a(b + c)=ab+ac\), we get \(4\times6t+4\times1 = 24t + 4\). Wait, no, wait, the original expressions: Wait, the two expressions are \(4(6t + 1)\) and \(24t+1\)? Wait, no, let's check the problem again. Wait, the user's problem: Thuy is substituting \(t = 3\) and \(t = 8\) to determine if two expressions are equivalent. The expressions are \(4(6t + 1)\) and \(24t + 1\)? Wait, no, maybe a typo? Wait, no, let's compute for \(t = 3\) first.

First, compute \(4(6t + 1)\) when \(t = 3\):
\(4(6\times3 + 1)=4(18 + 1)=4\times19 = 76\)

Then compute \(24t + 1\) when \(t = 3\):
\(24\times3 + 1=72 + 1 = 73\)

Now for \(t = 8\):
\(4(6\times8 + 1)=4(48 + 1)=4\times49 = 196\)

\(24\times8 + 1=192 + 1 = 193\)

So the two expressions give different results for both \(t = 3\) and \(t = 8\). Wait, but the options: Let's check the options again. The options are:

• Both expressions are equivalent to 73 when \(t = 3\). But we saw \(4(6t + 1)\) at \(t = 3\) is 76, \(24t + 1\) is 73. So no.

• Both expressions are equivalent to 76 when \(t = 3\). \(24t + 1\) at \(t = 3\) is 73, so no.

• Both expressions are equivalent to 193 when \(t = 8\). \(4(6t + 1)\) at \(t = 8\) is 196, \(24t + 1\) is 193. So no.

• The expressions are not equivalent. Which matches our calculation. Wait, but let's re - check the simplification of \(4(6t + 1)\). \(4(6t+1)=24t + 4\), not \(24t + 1\). So the two expressions \(24t + 4\) and \(24t + 1\) differ by 3, so they are not equivalent. So when \(t = 3\), \(24t + 4=76\), \(24t + 1 = 73\); when \(t = 8\), \(24t + 4 = 196\), \(24t + 1=193\). So the correct option is "The expressions are not equivalent."

Wait, but let's re - evaluate the options. The options are:

1. Both expressions are equivalent to 73 when \(t = 3\). (False, as \(4(6t + 1)=76\) at \(t = 3\))

1. Both expressions are equivalent to 76 when \(t = 3\). (False, \(24t + 1 = 73\) at \(t = 3\))

1. Both expressions are equivalent to 193 when \(t = 8\). (False, \(4(6t + 1)=196\) at \(t = 8\))

1. The expressions are not equivalent. (True, since they give different values for \(t = 3\) and \(t = 8\))

Answer:

The expressions are not equivalent. (The option "The expressions are not equivalent.")