SOLUTION: Marie is 4 years younger than Peter. if the product of their ages is 5 years ago is 45.What are their age now?

Question 497500: Marie is 4 years younger than Peter. if the product of their ages is 5 years ago is 45.What are their age now?
Answer by lmeeks54(111) (Show Source): You can put this solution on YOUR website!
You gotta' love two equations and two unknowns, no matter how the problem is worded. So, decompose the word problem into its two equations.
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Let's let P = Peter's age now
Let's let M = Marie's age now
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Marie is 4 yrs younger than Peter:
M = P - 4
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The product of their ages 5 yrs ago is 45
(P - 5) x (M - 5) = 45
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The basic method is to always be able to eliminate one of the unknowns so we can solve for that. That usually means re-casting one of the equations so that it is expressed in terms of one of the unknowns. But we already have that:
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M = P - 4
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So, let's plug this equality for M into our 2nd equation so that it is only expressed in terms of P:
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(P - 5) x ((P - 4) - 5) = 45
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Let's group our terms and simplify:
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(P - 5) x (P - 9) = 45
P^2 - 14P + 45 = 45
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with 45 on both sides of the "=" sign, we can substract 45 from both sides:
P^2 - 14P = 0
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We need to reduce the P^2 term. We can do that by rearranging the terms (adding 14P to both sides):
P^2 = 14P
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Now we can divide both sides by P to simplify further:
P = 14
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So now we know Peter's age today. We need to go back into one of the original equations to determine Marie's age today and we need to check our work with the equation about their ages 5 years ago:
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M = P - 4
M = (14) - 4
M = 10
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So, Marie is 10 years old today.
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Let's check our work with the 2nd equation:
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(P - 5) x (M - 5) = 45
(14 - 5) x (10 - 5) = 45
(9) x (5) = 45
45 = 45 is true, our answers check.
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P = 14 and M = 10 is the solution to our system of two equations.
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Cheers,
Lee
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