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Solving Linear and Quadratic Systems

May 9, 2025

Miss Smith's Math Tutorial: Systems of Linear and Quadratic Equations

Overview

  • Discussion on solving systems of equations involving one linear and one quadratic equation.
  • Methods: Algebraic solution, Graphical solution, and Calculator use.

System of Equations

  • A system consists of two equations:
    • Linear Equation: Highest power of variable = 1 (e.g., y = 4x + 1)
    • Quadratic Equation: Highest power of variable = 2 (e.g., y = x² + 4)
  • Equating two expressions for y gives us the system.

Algebraic Method

  1. Set Equations Equal: 4x + 1 = x² + 4
  2. Reorganize to Standard Form:
    • Move all terms to one side: x² - 4x + 3 = 0
  3. Identify a, b, c: a = 1, b = -4, c = 3
  4. Solve Using Quadratic Formula:
    • Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
    • Simplify step by step:
      • ( x = \frac{4 \pm \sqrt{16 - 12}}{2} )
      • ( x = \frac{4 \pm 2}{2} )
    • Solutions for x: 1 and 3
  5. Find Corresponding y Values:
    • Plug x = 1 into y = 4x + 1: y = 5 (Point: (1, 5))
    • Plug x = 3 into y = 4x + 1: y = 13 (Point: (3, 13))

Graphical Method

  • Graphing Linear Equation:
    • y = 4x + 1: Slope-intercept form (m = 4, b = 1)
    • Plot starting at y-intercept (1) and follow slope.
  • Graphing Quadratic Equation:
    • Plot using a graphing calculator or table.
    • Identify points including vertex and points around it.
  • Intersection Points:
    • Visualize the intersection of the line and parabola on the graph.

Calculator Method

  1. Use Graphing Calculator:
    • Clear previous equations.
    • Input equations: y = 4x + 1 and y = x² + 4
    • Graph both equations.
  2. Find Intersection Points:
    • Use 2nd -> Trace -> 5 to find intersections.
    • Confirm intersections at (1, 5) and (3, 13).

Notes on Solutions

  • Systems can have:
    • Two Solutions: Intersect at two points (as shown in this example).
    • One Solution: Line touches parabola at one point.
    • No Solution: Line does not intersect the parabola.

Practice Problem

  • Solve the system:
    • Linear Equation: y = 3x + 1
    • Quadratic Equation: y = -x² + 4x + 1
  • Determine intersecting points using any method.

Answers will be shared in the video description.

This tutorial covers a comprehensive approach to solving systems of equations and understanding their graphical representation. Miss Smith emphasizes the importance of understanding each method for different scenarios.

Full transcript