SOLUTION: The product of two positive integers is 98. If they differ by 7, what are the integer? Please help.

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
Answer by bucky(2189) (Show Source): 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:
.
x*y = 98
.
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:
.
x - 7 = y
.
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:
.
x*(x - 7) = 98
.
Multiply out the left side to get:
.
x^2 - 7x = 98
.
Subtract 98 from both sides of this equation and you get:
.
x^2 - 7x - 98 = 0
.
This equation can be factored to give:
.
(x - 14)(x + 7) = 0
.
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.
.
So set the two factors equal to zero and solve for the value of x that makes each factor
equal zero:
.
x - 14 = 0
.
Add 14 to both sides and you get x = +14
.
Then do the second factor:
.
x + 7 = 0
.
Subtract 7 from both sides to get x = -7
.
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.
.
Now we can go back to the first equation ... the equation says that the product of the two
integers is 98:
.
x*y = 98
.
But x is +14. Substituting this into the equation results in:
.
14*y = 98
.
Divide both sides of this equation by 14 to solve for y and you get:
.
y = 98/14 = +7
.
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.
.
Hope this helps you to understand the problem and how to solve it.
.

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