Question 102682: The product of two positive integers is 98. If they differ by 7, what are the integer?
Please help. Found 2 solutions by Earlsdon, bucky:Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website! Let the integers be x and y, then: and Rewrite this as and substitute into the first equation. Simplify. Subtract 98 from both sides. Solve this quadratic equation by factoring.
Then x = 7
y = x+7
y = 7+7
y = 14
The two integers are: 7 and 14
Check:
7*14 = 98
14-7 = 7
You can put this solution on YOUR website! Let one of the integers be x and the other integer by y
.
The product of the integers is 98. That means that x times y equals 98 and in equation form
this is:
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x*y = 98
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The integers differ by 7. This means that if we subtract 7 from one of the integers it
will equal the other integer. So we can say that:
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x - 7 = y
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This tells us that y = x - 7 so in the first equation we can substitute x - 7 for y. This
makes that product equation become:
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x*(x - 7) = 98
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Multiply out the left side to get:
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x^2 - 7x = 98
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Subtract 98 from both sides of this equation and you get:
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x^2 - 7x - 98 = 0
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This equation can be factored to give:
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(x - 14)(x + 7) = 0
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This equation will be true if either of the two factors equals zero ... because multiplication
by a zero on the left side makes the left side equal the zero on the right side.
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So set the two factors equal to zero and solve for the value of x that makes each factor
equal zero:
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x - 14 = 0
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Add 14 to both sides and you get x = +14
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Then do the second factor:
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x + 7 = 0
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Subtract 7 from both sides to get x = -7
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But x can't be -7 because the problem says the integers are positive. Therefore, the
only valid solution for x is that x = +14.
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Now we can go back to the first equation ... the equation says that the product of the two
integers is 98:
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x*y = 98
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But x is +14. Substituting this into the equation results in:
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14*y = 98
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Divide both sides of this equation by 14 to solve for y and you get:
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y = 98/14 = +7
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So the two integers (x and y) are 14 and 7. Their product is 98 and their difference
is 14 - 7 = 7 ... just as the problem required.
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Hope this helps you to understand the problem and how to solve it.
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