Question

How do you solve `y=-x^2+4, y=1/2x+5` using substitution?

Answer 1

`"no solution"`

Explanation:

`color(red)(y)=-x^2+4to(1)`

`color(red)(y)=1/2x+5to(2)`

`"since both equations express y in terms of x we can"`
`"equate them"`

`rArr1/2x+5=-x^2+4`

`"rearrange and equate to zero"`

`rArrx^2+1/2x+1=0`

`"check the nature of the roots using the "color(blue)"discriminant"`

`Delta=b^2-4acrarr(a=1,b=1/2,c=1)`

`color(white)(xx)=1/4-4<0`

`"hence there are no real solutions"`

`"that is the line and parabola do not intersect"`
graph{(y+x^2-4)(y-1/2x-5)=0 [-20, 20, -10, 10]}