We solve the 2 equations simultaneously by substituting the expression

`y = x -1`

into the expression

`x^2+y^2-x-3y=0`

[See some background to this at: Algebraic Solution of Systems of Equations.]

We have:

`x^2+(x-1)^2-x-3(x-1)=0`

`x^2+x^2-2x+1-x-3x+3=0`

`2x^2-6x+4=0`

`x^2-3x+2=0`

`(x-1)(x-2)=0`

So we see that the solutions for x are `x = 1` or `x = 2`. This gives the corresponding y-values of `y = 0` and `y = 1`.

So the points of intersection are at: `(1, 0)` and `(2, 1)`.

We can see that our answer is correct in a sketch of the situation:

simultaneous solution - line intersecting circle

Exercise for You

Where did (0.5, 1.5) for the center of the circle come from?

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