SOLUTION: What two numbers have the sum of 9 and the product of 20?

Question 92741: What two numbers have the sum of 9 and the product of 20?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Call the two numbers x and y
.
Since their sum is 9 we write this in equation form as:
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x + y = 9
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And since their product is 20 we write this in equation form as:
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x*y = 20
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We now have two equations. One way to solve an equation set such at this is by using substitution.
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Solve the first equation regarding the sum for one of the variables in terms of the other.
You can do this by subtracting y from both sides to get:
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x + y - y = 9 - y
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Perform the subtraction on the left side and the equation reduces to:
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x = 9 - y
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Then in the second equation we can substitute 9 - y for x since x is equal to 9 - y.
After making this substitution the second equation becomes:
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(9 - y)*y = 20
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Doing the distributed multiplication on the left side (multiplying the y by each of the terms
in the set of parentheses) results in the equation becoming:
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9y - y^2 = 20
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rearrange the terms on the left side in descending powers of y to get:
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-y^2 + 9y = 20
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Subtract 20 from both sides and the equation becomes:
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-y^2 + 9y - 20 = 0
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Finally, to get things in a little more standard form, multiply all the terms on both sides
by -1 to make the y^2 term be positive. After this multiplication, the equation becomes:
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y^2 - 9y + 20 = 0
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This factors into:
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(y - 5)*(y - 4) = 0
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This equation will be true if either one of the factors equals zero, because a multiplication
by zero on the left side will result in that side becoming zero and therefore it equals the
right side.
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So set y - 5 = to 0. Solve for y by adding 5 to both sides to get y = 5.
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Then set y - 4 equal to 0. Solve for y by adding 4 to both sides to get y = 4.
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So we have two possibilities ... either y = 5 or y = 4.
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Now go to the equation that said x + y = 9. If y is equal to 5 this equation becomes
x + 5 = 9 and when you subtract 5 from both sides this tells you that x = 4. However,
if y is equal to 4 this equation becomes x + 4 = 9 and when you subtract 4 from both sides
you get that x = 5.
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So the answer is that the two numbers are 4 and 5. Their sum is 9 and their product is 20.
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Hope this helps you to see your way through this problem.
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