# Where do the graphs of x+y=65 and x^2+y^2=194 cross (if they cross)?

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### 1 Answer

We have the equations

1) `x + y = 65`

2) `x^2 + y^2 = 194`

Solve the first equation for y:

`y = 65 - x`

Substituting into the second equation we obtain

`x^2 + (65-x)^2 = 194`

` `which implies that

`x^2 + 65^2 - 2(65)x + x^2 - 194 = 0`

`implies` `2x^2 -130x + 4031 = 0`

Using the quadratic formula we find that

`x = (130 pm sqrt(130^2 - 4(2)(4031)))/4 = 130/4 pm sqrt(-15348)/4`

Therefore there is no *Real* solution for `x`.

To check this look at the plotted graphs

**The graphs - a line y = 65 - x and a circle, center (0,0) radius sqrt(194) - **

**don't cross. Therefore the system of equations has no Real solutions.**

**The complex (conjugate) solutions satisfy**

**x = 130/4 +/- sqrt(-15348)/4**