Quadratic-Linear Systems Overview

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Solve the linear equation for y.

x + 1 = -x^2 + 6x - 3

x^2 - 5x + 4 = 0 Signup and view all the answers

(x - 4)(x - 1) = 0 Signup and view all the answers

x = 4 and x = 1 Signup and view all the answers

Plug x into either equation to find y. Signup and view all the answers

{(4,5),(1,2)} Signup and view all the answers

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A consistent system is one that has at least one solution, meaning the graphs of the equations intersect at some point.

Step 1: Solve the Linear Equation for y
- Example: From (y - x = 1), rearranging gives (y = x + 1).
Step 2: Substitute y into the Quadratic Equation
- Replace (y) in the quadratic (y = -x^2 + 6x - 3) with the linear equation’s expression, resulting in: (x + 1 = -x^2 + 6x - 3).
Step 3: Write the Quadratic in Standard Form
- Rearranging leads to: (x^2 - 5x + 4 = 0).
Step 4: Solve for x
- Factoring the quadratic gives ((x - 4)(x - 1) = 0), leading to:
  - (x = 4)
  - (x = 1)
Step 5: Plug x into Either Equation to Find y
- Using the respective (x) values:
  - For (x = 4): (y - 4 = 1) yields (y = 5).
  - For (x = 1): (y - 1 = 1) gives (y = 2).
Step 6: Write as Coordinates
- The solutions represented as coordinates are: ((4, 5)) and ((1, 2)).