Question

For what values of y does the following equations have no solutions for x

xy+ 5y+4x+20 = 1

Answer 1

y = -4

Explanation:

If we look at xy + 5y, we see that both terms share a factor of y. This gives us

y(x + 5) + 4x + 20 = 1

Similarly, 4x and 20 share a factor of 4, giving us

y(x + 5) + 4(x + 5) = 1

Now we see that these two terms share a factor of x + 5, so factoring gives us

(y + 4)(x + 5) = 1

Dividing both sides by y+4 gives us

x + 5 = 1/(y + 4)

Finally, we subtract both sides by 5 to get

x = 1/(y + 4) - 5

Thus, since we can not have a 0 in the denominator, we see that if y = -4 then x will have no solution. For any other value of y, we can find the solution for x using this equation.

Answer 2

y = - 4

collect terms in x

xy + 4x = 1 - 20 = - 19 ( factor out x )

x(y + 4 ) = - 19 ( divide both sides by ( y + 4) )

x =
`(-19)/(y + 4)`

x will be undefined if the denominator of the fraction is zero

y + 4 = 0 ⇒ y = - 4

x has no solution for y = - 4