System of equations

Question

(X-3)^2+(y+2)^2=16

2x+2y=10

Answer 1

The given equations are a circle and a line. A line will intersect a circle at two points if it passes through it, at one point if it is tangent to the circle, or at zero points if it never intersects the circle.

Solve the second equation for x:

2x + 2y = 10

Subtract 2y from both sides:

2x = -2y + 10

Divide both sides by 2:

x = -y + 5

Plug that expression in for x in the first equation:

(x - 3)² + (y + 2)² = 16

(-y + 5 - 3)² + (y + 2)² = 16

(-y + 2)² + (y + 2)² = 16

FOIL both binomials:

y² - 4y + 4 + y² + 4y + 4 = 16

Combine like terms:

2y² + 8 = 16

Subtract 8 from both sides:

2y² = 8

Divide both sides by 2:

y² = 4

Take the square root of both sides. Remember that the square root of a positive number can be positive or negative:

y = ±2

Plug both of those answers into the simplified equation x = -y + 5:

x = -(2) + 5

x = 3

x = -(-2) + 5

x = 7

So the two solutions are:

(3, 2) and (7, -2)