Question

What are the solutions to the equation x-7/x=6?

a. x=-7 and x=1
b. x=-6 and x=-1
c. x=-1 and x=7
d. x=1 and x=6

Answer 1

C

Explanation:

The equation is
`x-(7)/(x)=6`

LCM of the denominator on the left side gives us:

`(x^(2)-7)/(x)=6`

Cross multiplying and arranging gives us:

`x^(2)-7=6x\\x^(2)-6x-7=0`

This is a binomial, to factor this, we have to think of 2 numbers that multiplied gives us
`-7` [the constant term] and added gives us
`-6` [the coefficient of x].

The two numbers are:
`-7` and
`1`.

Now we can write:

`x^(2)-6x-7=0\\(x-7)(x+1)=0`

Hence, either
`(x-7)=0` OR
`(x+1)=0`. This gives us two solutions,
`x=7` and
`x=-1`. Answer choice C is correct.

Answer 2

`x - (7)/(x) = 6`

`(x^(2))/(x) - (7)/(x) = 6`

`(x^(2) - 7)/(x) = 6`

`x^(2) - 7 = 6x`

`x^(2) - 6x - 7 = 0`

`x^(2) - 7x + x - 7 = 0`

`x(x) - x(7) + 1(x) - 1(7) = 0`

`x(x - 7) + 1(x - 7) = 0`

`(x + 1)(x - 7) = 0`

`x + 1 = 0`
`or`
`x - 7 = 0`

`x = -1`
`or`
`x = 7`