Block 2 Systems of Equations and Inequalities Notes

Math 20-1 Unit 3 - Systems of Equations and Inequalities

Overview

  • Focus on Systems of Equations and Inequalities.
    • Chapter 8: Linear and Quadratic Inequalities
    • Chapter 9: Detailed lessons and examples.

Lesson Plan

  1. Solving Systems of Equations Graphically (8.1)

    • Homework: Page 435, Problems #1-3, 4-5 (only a-c), 8

  2. Solving Systems of Equations Algebraically (8.2)

    • Homework: Page 451, Problems #1, 2, 3-5 (only a and c), 8, 9

  3. Linear Inequalities in Two Variables (9.1)

    • Homework: Page 472, Problems #1-3 (only a and c), 4-5 (only a and c), 8ac, 9, 11

  4. Quadratic Inequalities in One Variable (9.2)

    • Homework: Page 484, Problems #1, 2, 3, 4-9 (only a and c)

  5. Quadratic Inequalities in Two Variables (9.3)

    • Homework: Page 496, Problems #1, 2 (a and c only), 3, 4-7 (a and c only), 8

Lesson 1: Solving Systems Graphically

  • Definition: An ordered pair (x,y) that satisfies both equations is a solution.
    • Example: For the system:
    • 2y = x + 2
    • y = x,
    • The point (2,4) satisfies both if it fits the equations.
  • Types of Solutions:
    • No real solution
    • One real solution
    • Two real solutions
    • Infinite solutions (quadratic-quadratic systems)

Lesson 2: Solving Systems of Equations Algebraically

  • Methods to Solve:
    • Graphically
    • Substitution
    • Elimination
  • Example:
    • Given equations: 5x - y = -10 and 2x - 2y = -4, use substitution.
      • Solve for x and y and verify the solution.

Lesson 3: Linear Inequalities in Two Variables

  • Forms of Linear Inequalities:
    • Ax + By < C
    • Ax + By ≤ C
    • Note the significance of the line (solid or dashed).
  • Solution Region:
    • All points satisfying the inequality in the Cartesian Plane.
  • Graphing Techniques:
    1. Rewrite in slope-intercept form: y = mx + b
    2. Identify the line type (solid for ≤, ≥; dashed for
    3. Shade the appropriate region based on inequalities.

Lesson 4: Quadratic Inequalities in One Variable

  • Forms of Quadratic Inequalities:
    • ax^2 + bx + c < 0
    • ax^2 + bx + c > 0
    • ax^2 + bx + c ≤ 0
    • ax^2 + bx + c ≥ 0
  • Solution Sets:
    • Can have no values, one value, or infinite values.
  • Example Solving:
    1. Solve 2x^2 - 3x - 2 ≤ 0.
    2. Use graphical method to find intersections and verify.

Lesson 5: Quadratic Inequalities in Two Variables

  • Quadratic inequalities: y < ax^2 + bx + c.
    • Parabola as the boundary, representing solutions in the Cartesian plane.
  • Graphing Example:
    • Determine whether a point (e.g. (2, -4)) is a solution to the given inequality.
  • Application of Inequalities:
    • Various practical problems modeled by quadratic equations.

Review Questions

  1. What is the x-coordinate of the intersection point for given equations?
  2. Find the sum of the x-coordinates from all intersections.
  3. How many solutions does the quadratic-quadratic system have?
  4. Solve the quadratic equation derived from a system.
  5. Determine the value of k given the solutions to a system.
  6. Identify points not in the solution region of a specified inequality.
  7. Calculate b - c from a quadratic formed by solving a system.
  8. Identify points that satisfy simultaneous inequalities.