We recognize that this is a straight line intersecting a circle. (See more on the circle.)

We may have:

We can simply substitute the right hand side of the first equation into the second equation:

x2 + (x + 1)2 = 25

This gives:

x2 + x2 + 2x + 1 = 25

2x2 + 2x − 24 = 0

x2 + x − 12 = 0

(x + 4)(x − 3) = 0

So `x = −4` or `x = 3`.

This gives our intersecting points to be: `(−4, −3)` and `(3, 4)`.

Is it correct? The graph showing the line intersecting the circle is as follows:

123456-1-2-3-4-5-6123456-1-2-3-4-5-6xyOpen image in a new page

Graphs of `y = x+1` and `x^2+y^2=25`: Intersection line and circle

We can see from the graph that our solutions `(−4, −3)` and `(3, 4)` are correct.

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