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The product of two numbers is 800.  Their difference is 20. What are the two numbers?

11/21/2021

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The product of two numbers is 800.  The difference between the numbers is 20.  What are the two numbers?

x = one number
y = the other number


xy = 800  {the product of two numbers is 800}
​x - y = 20  {their difference is 20}

Solve the second equation for x, by adding y to each side.
x - y = 20  {the second equation}
x = y + 20  {added y to each side}

Substitute (y + 20), in for x, into first equation.
xy = 800  {first equation}
(y + 20)y = 800  {substituted (y + 20) in for x}
y² + 20y = 800  {used distributive property}
y² + 20y - 800 = 0  {subtracted 800 from each side}
(y + 40)(y - 20) = 0  {factored into two binomials}
y + 40 = 0 or y - 20 = 0  {set each factor equal to zero}
y = -40 or y = 20  {solved each equation for y}

Substitute -40 and 20, back in for y, into first equation.
If y = -40
xy = 800  {first equation}
-40x = 800  {substituted -40 for y}
x = -20  {divided each side by -40}

The product of -20(-40) = 800
The difference of -20 - (-40) = -20 + 40 = 20  {this could be argued, based on the negatives}

​
If y = 20
xy = 800 {first equation}
20x = 800  {substituted 20 in for y}
x = 40  {divided each side by 20}
The product of 20(40) = 800
the difference of 40 - 20 = 20


x = 40 and y = 20
or 
x = -20 and y = -40


Pfew!

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