# Roger is 6 years older than Zoe.Three years ago,Zoe's age was two-thirds of Roger's age.Find their present age. Please try using the substitution or elimination method as i am currently learning...

Roger is 6 years older than Zoe.Three years ago,Zoe's age was two-thirds of Roger's age.Find their present age.

Please try using the substitution or elimination method as i am currently learning about that.Thanks

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### 1 Answer

Let x be the age of Roger.

Let y be the age of Zoe.

We know that AT PRESENT, x = y + 6.

Three years ago, the age of Roger is x - 3. The age of Zoe is y - 3.

We know that THREE YEARS AGO, `y - 3 = 2/3 (x- 3).`

Now, we have two equations. The first equation is the relationship of their ages at present: x - y = 6. While the second equation is the relationship of their ages three years ago (which simplifies to): **-2x + 3y = 3**.

Let us use substitution. Isolating x from the first equation, we get x = 6 + y. We then substitute this to the second equation:

`-2x + 3y = 3 rArr -2(y+6) + 3y = 3 rArr -2y - 12 + 3y = 3 rArr y - 12 = 3 rArr y = 15.`

This implies that x = 21 (since x - 15 = 6).

Therefore, at present, Roger is 21 years old and Zoe is 15 years old.

[To check: three years ago, Roger was 18, Zoe 12; and 12 is 2/3 of 18. Hence, the answer is correct]