Question

Andrea is 6 years younger than Tony. In two years, Tony will be three times as old as Andrea. Let a be Tony’s current age. Which of these linear equations can be used to find Andrea’s and Tony’s current ages?(1 point) Responses a+2=3(a−6+2) a plus 2 equals 3 left parenthesis a minus 6 plus 2 right parenthesis a+2=3(a+6+2) a plus 2 equals 3 left parenthesis a plus 6 plus 2 right parenthesis a=3(a+2) a equals 3 left parenthesis a plus 2 right parenthesis 3(a+2)=a−6+2 3 left parenthesis a plus 2 right parenthesis equals a minus 6 plus 2

Answer 1

To solve the problem, we first need to define Andrea’s age in terms of Tony’s age.

Let:
- \( a \) = Tony’s current age
- \( a - 6 \) = Andrea’s current age (since Andrea is 6 years younger than Tony)

In two years:
- Tony’s age will be \( a + 2 \)
- Andrea’s age will be \( (a - 6) + 2 = a - 4 \)

According to the problem, in two years, Tony will be three times as old as Andrea:

\[
a + 2 = 3(a - 4)
\]

Now, let's simplify this equation:

\[
a + 2 = 3a - 12
\]

Rearranging gives us:

\[
a - 3a = -12 - 2
\]
\[
-2a = -14
\]
\[
a = 7
\]

So, Tony's current age is 7 and Andrea's age is \( 7 - 6 = 1 \).

Now, let's check the response options given:

1. \( a + 2 = 3(a - 6 + 2) \)
- \( a + 2 = 3(a - 4) \) (this one is correct)

2. \( a + 2 = 3(a + 6 + 2) \)
- This does not represent the scenario correctly.

3. \( a = 3(a + 2) \)
- This does not represent the scenario correctly.

4. \( 3(a + 2) = a - 6 + 2 \)
- This simplifies to \( 3(a + 2) = a - 4 \) (not correct)

The correct equation that can be used to find Andrea’s and Tony’s current ages is:

**\( a + 2 = 3(a - 4) \)**, which corresponds to the first option you provided:

**\( a + 2 = 3(a - 6 + 2) \)** or simply the rearranged version \( a + 2 = 3(a - 4) \).