# How does one restate this mathematically?

This isn't my homework. It's a math problem I picked up while reading a book to promote clearer thinking. I know the answers are 56 and 42, but I'm wondering about how to restate the second equation in the system with this problem: 'the ages of a man and his wife together are 98. He is twice as old as she was when he was the age she is today. What are their ages?'

Obviously, the first equation would be $x + y = 98$. Does one state the second equation as $x/2 = y - z$? But then wouldn't that add another variable?

Thank you.

Note by Michael David Sy
4 years, 10 months ago

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I suppose you meant to say "He is as old as 's'he was ...."

You can take the variables as the current age of the wife, $$y$$and the difference between their ages, $$a$$

Then the man's current age is $$y+a$$

When the man was as old as his wife is today, he was $$y$$ years old and his wife was $$y - a$$

We are given that $$y +a = 2(y - a)$$

Hence the two equations.

- 4 years, 10 months ago

That still creates a third equation, because a is not known. Thank you! :)

- 4 years, 10 months ago

Yes, but that will lead to $$a$$ being the second variable and not $$x$$ One equation will be$$y + a = 2(y-1)$$ The second is $$y + (y + a) = 98$$

- 4 years, 10 months ago

Thanks. It really confused me. :)

- 4 years, 10 months ago

?? Wouldn't the second EQ just be $$x = 2y?$$

- 4 years, 10 months ago