How many real solutions does this system of equations have? x^2 + y^2 = 36
3x – y + 1 = 0
Answer choices:
A.0
B.1
C.2
D.3

C:  there are two real solutions

Step-by-step explanation:

x^2 + y^2 = 36 is the equation of a circle centered at the origin and with radius √36, or 6.  Even a rough sketch of this circle would enable you to visualize what is happening here.  

3x – y + 1 = 0 may be solved for y:  y = 3x + 1.  The y-intercept of this straight line is (0, 1).  Plot this point inside the circle of radius 6 mentioned above and then draw a straight line with slope m = 3 through it.  This line will intersect the circle in two places; they represent real solutions.

Alternatively, substitute 3x + 1 for y in x^2 + y^2 = 36:

x^2 + (3x + 1)^2 = 36, or

x^2 + 9x^2 + 6x + 1 = 36, or

10 x^2 + 6x + 1 - 36 = 0, or

10 x^2 + 6x - 35 = 0.  This is a quadratic equation.  We'll use the quadratic formula to find solutions which represent the intersections of this line with this circle:

The coefficients are a = 10, b = 6 and c = -35.  Thus, the discriminant is

b^2 - 4ac, or 6^2 - 4(10)(-35), or 1436.  

Because the discriminant is positive, we can safely conclude that there are two real solutions, that is, two different, real x-values, each representing the x-coordinate of a point of intersection of the circle and the line.

The correct answer to this prolem is C:  there are two real solutions.

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