Question

swati is substituting ( t = 5 ) and ( t = 9 ) to determine if the two expressions are equivalent. ( 8(4t - 3) ) ( 32t - 24 ) which statement is true? both expressions are equivalent to 264 when ( t = 9 ). both expressions are equivalent to 157 when ( t = 5 ). both expressions are equivalent to 46 when ( t = 5 ). the expressions are not equivalent.

Explanation:

Step1: Simplify \(8(4t - 3)\)

Using the distributive property \(a(b - c)=ab - ac\), we have \(8\times4t-8\times3 = 32t - 24\). So the two expressions \(8(4t - 3)\) and \(32t - 24\) are algebraically equivalent. Now we can check the values for \(t = 5\) and \(t=9\) (though since they are algebraically equivalent, their values should be equal for any \(t\)).
For \(t = 5\):

• For \(8(4t - 3)\): Substitute \(t = 5\), we get \(8(4\times5 - 3)=8(20 - 3)=8\times17 = 136\)? Wait, no, wait, let's recalculate. Wait, \(4t-3\) when \(t = 5\) is \(4\times5-3=20 - 3 = 17\), then \(8\times17 = 136\)? But the other expression \(32t-24\) when \(t = 5\) is \(32\times5-24=160 - 24 = 136\). Wait, the options have some different numbers. Wait, maybe I misread the options. Wait the options:

Wait the problem says "Swati is substituting \(t = 5\) and \(t = g\) to determine if the two expressions are equivalent. The expressions are \(8(4t - 3)\) and \(32t - 24\). Which statement is true?

Options:

• Both expressions are equivalent to 264 when \(t = 9\).

• Both expressions are equivalent to 157 when \(t = 5\).

• Both expressions are equivalent to 46 when \(t = 5\).

• The expressions are not equivalent.

Wait let's check \(t = 5\) for \(8(4t - 3)\):

\(8(4\times5 - 3)=8(20 - 3)=8\times17 = 136\).

For \(32t - 24\) when \(t = 5\): \(32\times5-24 = 160 - 24 = 136\). So the first three options have wrong values for \(t = 5\). Now check \(t = 9\):

For \(8(4t - 3)\): \(8(4\times9 - 3)=8(36 - 3)=8\times33 = 264\).

For \(32t - 24\) when \(t = 9\): \(32\times9-24=288 - 24 = 264\). So when \(t = 9\), both expressions equal 264. So the first option is correct.

Step2: Verify for \(t = 9\)

Calculate \(8(4t - 3)\) at \(t = 9\): \(8(4\times9 - 3)=8(36 - 3)=8\times33 = 264\).

Calculate \(32t - 24\) at \(t = 9\): \(32\times9-24 = 288 - 24 = 264\).

Since both expressions give 264 when \(t = 9\), and we already saw they are algebraically equivalent (since \(8(4t - 3)=32t - 24\) by distributive property), the first statement is true.

Answer:

Both expressions are equivalent to 264 when \(t = 9\).