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# How do I solve a pair of simultaneous equations where one is linear and the other is a circle?

By Kris Boulton on the 11th of June, 2012

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If one equation is linear and the other is the equation of a circle then you need to use the method of elimination.

The linear equation should be of the form y=mx+c.  You need to substitute (mx+c) into your second equation instead of y. Then rearrange it, collect alike terms and solve.

By Lee Mansfield on the 13th of June, 2012

## Solve a pair of simultaneous equations where one is linear and the other is a circle

A circle equation has the form ax2 + by2 = r2, where r is the radius of the circle.

These can only be solved algebraically using the method of substitution.

Graphically, the solutions for x and y are given by the coordinates at which the straight line graph cuts the circle.

Method of Substiution:

1) Rearrange the pair of simultaneous linear equations into terms of x or y. So y = .. or x = ..
2) Substitute this expression into the circle equation, so that you now have an equation in terms of one variable, and you can solve it.
Remember: in the equation of a circle you have y2 and x2, so you must square what you're substiuting in!
3) Plug your answer back into one of the equation, and solve the equation for the other variable.
4) Check your solution with the other equation.

For example:

x^2 + y^2 =25.....(1)
y=x+1......(2)

x^2 + (x+1)^2 = 25

x^2 + x^2 +2x +1 =25
-25 -25
2x^2 +2x-24 = 0
x^2 + x -12 =0
(x-3) (x+4) = 0
so x= 3 or x=4

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