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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=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 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

1 answer

Let's break down the problem:

  1. Let \( a \) be Tony's current age.
  2. Since Andrea is 6 years younger than Tony, her current age is \( a - 6 \).
  3. In two years, Tony's age will be \( a + 2 \) and 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, which gives us the equation:

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

Now, let's manipulate this equation:

  1. Expand the right side: \[ a + 2 = 3a - 12 \]

  2. Rearranging gives: \[ 2 + 12 = 3a - a \] \[ 14 = 2a \] \[ a = 7 \]

So, the equation we derived that relates Tony’s and Andrea's ages is \( a + 2 = 3(a - 4) \).

Now, let's see which of the options fits:

  1. \( a = 3(a + 2) \) → Incorrect
  2. \( 3(a + 2) = a - 6 + 2 \) → Incorrect
  3. \( a + 2 = 3(a - 6 + 2) \) → Correct (this is equivalent to our equation)
  4. \( a + 2 = 3(a + 6 + 2) \) → Incorrect

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

Option 3: \( a + 2 = 3(a - 6 + 2) \)