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

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mathsworkmusic | (Level 2) Educator

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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

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