The circle x^2 + y^2 = 25 and the line y = 2x - 5 intersect at two points. What is the sum… — ACT

The circle $x^2 + y^2 = 25$ and the line $y = 2x - 5$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

Answer: -2

Answer

The sum of the

y

-coordinates of the intersection points is

-2

Substituting

y = 2x - 5

into the circular equation

x^2 + y^2 = 25

yields the quadratic equation

5x^2 - 20x = 0

. Factoring this equation gives

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. The sum of these coordinates is

-5 + 3 = -2

Step-by-Step Solution

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

x^2 + (2x - 5)^2 = 25

To find the points of intersection, we solve the system of equations by substitution.

Expand and simplify the resulting equation.

5x^2 - 20x = 0

Expanding

(2x - 5)^2

gives

4x^2 - 20x + 25

. Combining like terms and subtracting 25 from both sides simplifies the equation.

Factor the quadratic equation to solve for

x

x = 0

x = 4

Factoring out

5x

gives

5x(x - 4) = 0

, which yields the roots

x = 0

and

x = 4

Find the corresponding

y

-coordinates by substituting the

x

-values back into

y = 2x - 5

The intersection points are

(0, -5)

and

(4, 3)

For

x = 0

y = 2(0) - 5 = -5

. For

x = 4

y = 2(4) - 5 = 3

Calculate the sum of the

y

-coordinates.

-2

Adding the

y

-coordinates

-5

and

3

gives

-5 + 3 = -2

Key Concept

Systems of Linear and Non-Linear Equations

Question

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