U6 Concept Review-Systems

Key Ideas

Solving Systems of Equations Graphically

  • Linear-Quadratic Equations:

    • Isolate y in each equation.

    • Graph the line and parabola on the same grid.

    • Solutions are points of intersection (x, y).

    • Verify solutions in original equations.

    • Three possibilities for intersections and solutions.

  • Quadratic-Quadratic Equations:

    • Isolate y for both parabolas.

    • Graph both parabolas.

    • Solutions are points of intersection (x, y).

    • Verify solutions in original equations.

    • Infinite solutions if one quadratic is a multiple of another.

Solving Systems Algebraically

  • Methods:

    • Substitution: Solve one equation for one variable, then substitute it into the other equation.

    • Elimination: Align terms, eliminate one variable, and solve for the other.

Example Problems

Linear-Quadratic Example

  • Given:

    1. y = x² + x

    2. x - y = -3

  • Use elimination method:

    • Align and manipulate equations.

    • Resulting equation: 0 = x² + 2x - 3.

    • Factor: (x + 3)(x - 1) = 0

    • Solutions: x = -3 or x = 1.

  • Substitute back to find y:

    • For x = -3: y = 0 (Solution: (-3, 0))

    • For x = 1: y = 4 (Solution: (1, 4)).

Quadratic-Quadratic Example

  • Given:

    1. y = 2x² - 2x + 3

    2. y = x² + 5x - 7

  • Substitution or elimination:

    • Resulting equation: x² - 7x + 10 = 0 => (x - 5)(x - 2) = 0.

  • Solutions: x = 5 or x = 2.

  • Substitute back to find y:

    • For x = 5: y = 43

    • For x = 2: y = 7.

  • Solutions: (5, 43) and (2, 7).

Further Quadratic Example

  • Given:

    1. 6x² - x - y = -1

    2. 4x² - 4x - y = -6

  • Use elimination:

    • Resulting equation: 2x² + 3x = 5.

  • Solve for x, get x = -5 or x = 1, find y for both.

  • Solutions: (1, 6) and (5, 41).

Vocabulary

  • System of Linear-Quadratic Equations: A linear equation and a quadratic equation in the same variables (line and parabola).

  • System of Quadratic-Quadratic Equations: Two quadratic equations in the same variables (two parabolas).

  • Solution: Points (x, y) satisfying all equations; intersections of graph.

  • Method of Substitution: Solve one equation for a variable, substitute into the other.

  • Method of Elimination: Add/subtract equations to eliminate a variable.

Common Errors

  • Isolating a Variable: Errors with signs.

  • Subtraction Mistakes: Only subtracting first term.

  • Not Checking Solutions: Failing to verify solutions in original equations.