Question

One integer is 7 less than 5 times another. Their product is 24. Find the integers.

Answer 1

3, 8

Explanation:

Let's make the first integer x and the second integer y.

Now, you take the information given and convert it into a system of equations:

xy = 24

x = 5y - 7

y = 24 / x

Now, we substitute:

(5y - 7) · y = 24

Distribute y

5y² - 7y = 24

Rearrange it into a quadratic equation:

ax² + bx + c = 0

5y² - y - 24 = 0

Use the quadratic formula (shown in image) and plug in the values:

x = (- (7) ±
`√((-7^))^(2) -4(5)(-24)}` ) / 2(5)

`\sqrt{(-7)^(2) -4(5)(-24)}` =
`√(49 - (-480)) = √(529) = 23`

Remember that ± means there are two y values

`x = (7+23)/(10) = (30)/(10) = 3`

and

`x = (7-23)/(10) = (-16)/(10) = -1.6`

Since y must be an integer, we know that y = 3

Now, plug in y:

5(3) - y = 24

15 - y = 24

- y = -24

y = 8

Let's check to see if we are right:

5y - 7 = 5(3) - 7 = 15 - 7 = 8

It works!