Hi, there--

Problem: 
Bads is thinking of two numbers .one number is 7more than the other number .the product of 
the number is 60.what are the numbers?

Solution:
You can solve this a non-algebra way or an algebra way.

I. The non-algebra method.
Think of a number pair whose product is 60. Check to see if the difference is 7.
(1, 60) --> 1 * 60 = 60 ---> 60 - 1 = 59 
(2,30) --> 2 * 30 = 60 --> 30 - 2 = 28
(3, 20) --> 3 * 20 = 60 --> 20 - 3 = 17

Continue in this manner until you find the pair of numbers whose product is 60 and whose difference is 7.

(4, 15) --> 4 * 15 = 60 --> 15 - 4 = 11
(5, 12) --> 5 * 12 = 60 --> 12 - 5 = 7

Bingo! The two numbers are 5 and 12.

II. The algebra method
Let x be the smaller number.
Let y be the larger number.

The product of the two numbers is 60, so x * y = 60.

The difference between the numbers is 7, so y - x = 7.

Let's rewrite this second equation in "y=" form by adding x to both sides.
y - x = 7
y = 7 + x

Substitute 7 + x for y in the first equation.
x * y = 60
x * (7 + x) = 60

Solve for x. Clear the parentheses by using the distributive property.
7x + x^2 = 60

We have a quadratic equation. Subtract 60 from both sides and rearrange the terms in 
descending order.
x^2 + 7x - 60 = 0

Rewrite the equation in factored form. We want two factors with a product of -60, and a 
difference of 7. The factors are 12 and -5.

(x + 12)(x - 5) = 0

Solve.
x = -12 or x = 5

Try each solution in the real-world problem.

If x = 5, then the smaller number is 5 and the larger number is 12 (because 12 - 5 = 7).
The product of 5 and 12 is 60, so this pair works.

If x = -12, then the smaller number is -12 and the larger number is -5 
(because -12 - (-5) = 7). The product of -5 and -12 is 60, so this number pair also works.

So, you have two possible number pairs: -5 and -12 or 5 and 12. 

Hope this helps,

Mrs.Figgy
math.in.the.vortex@gmail.com