Intersection of a Line and a Parabola: Solving y5x-6 and yx^2 for Common Points

In this article, we will explore the intersection of a line and a parabola by examining the equations y5x-6 and yx^2. We will solve the problem step-by-step, highlighting the methods and techniques used in coordinate geometry.

Problem Statement

We are given two equations: y5x-6 and yx^2. Our goal is to find the coordinates of the points where the line y5x-6 intersects the parabola yx^2.

Step-by-Step Solution

Since the line and the parabola intersect, their y-coordinates must be equal. Therefore, we set the equations equal to each other:

1. Set the two equations equal:

y5x-6 and yx^2

2. Equate the two:

x^2 5x-6

3. Rearrange the equation to form a standard quadratic equation:

x^2 - 5x 6 0

4. Factor the quadratic equation:

x^2 - 3x - 2x 6 0

x(x - 3) - 2(x - 3) 0

(x - 3)(x - 2) 0

5. Solve for x:

x 3 or x 2

The x-coordinates of the points of intersection are 3 and 2.

Find the Corresponding y-Coordinates

To find the corresponding y-coordinates, we can substitute these x-values back into either equation. We will use the equation of the line y5x-6:

For x 2:

y 5(2) - 6

y 10 - 6

y 4

For x 3:

y 5(3) - 6

y 15 - 6

y 9

The points of intersection are thus (2, 4) and (3, 9).

Conclusion

By following the steps above, we have successfully determined the points of intersection between the line y5x-6 and the parabola yx^2. These points are (2, 4) and (3, 9).

Verification

To verify the solution, you can plot the line and the parabola on a graph and check for the points of intersection. You can use online graphing tools like Desmos to verify the solution.

Desmos Graphing Calculator

Feel free to explore more problems in coordinate geometry and challenge yourself with similar questions.

Key Concepts

Intersection of a Line and a Parabola: This involves solving a system of equations where one is linear and the other is quadratic. The solutions provide the coordinates of the points where these two functions meet. Solving Quadratic Equations: Understanding how to factor and solve quadratic equations is crucial. The quadratic formula can also be used, but factoring provides a clear insight into the problem. Coordinate Geometry: Utilizing the coordinate system to solve geometric problems algebraically is a powerful technique. It combines algebra and geometry, allowing us to solve complex problems graphically and analytically.