Is There a Simpler Method to Solve Similar Linear and Quadratic Systems?

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azure kitsune
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Hey everyone, today in physics class, my teacher reduced a physics problem to solving a system of two equations in this form:

ax+by=c
ax2+by2=c

Where a, b, and c are constants. Then my teacher solved for x and y by solving for x in the first equation, plugging that into the second equation, resulting a very intimidating quadratic in y, and solving for y using the quadratic formula.

I was wondering if there was an easier way to solve for x and y in this situation. I have a feeling there would be some kind of a shortcut because of the similarities in the two equations.
 

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No, I think solving for either x or y in the first equation and then putting the result into the second is the simplest way to solve those equations.