In-Depth Notes on Solving Equations and Graphical Methods(dec 28(pt.2)(1:8:43))

Understanding System of Equations

• Focus on manipulating equations to isolate variables.

• Example: Canceling variables to simplify equations.

• If given equations like:

  • 3x + y = 7

  • y - 2x = -4

• You can multiply one equation to facilitate elimination of the variable.

• Substitute found values back into one of the original equations to find the other variable.

Graphical Representation

• Drawing graphs helps visualize solutions, especially for quadratic equations.

• Recognize the standard form of an equation of a circle and line:

  • Circle: (x - h)^2 + (y - k)^2 = r^2

  • Line: y = mx + b, where $m$ is the slope and $b$ is the y-intercept.

• Example: Finding the intersection points helps understand real-life implications of these equations.

Types of Solutions in Quadratic Equations

• Using the discriminant ext{delta} = b^2 - 4ac determines the nature of solutions:

  • If ext{delta} > 0: Two distinct real solutions

  • If ext{delta} = 0: One real solution (repeated)

  • If ext{delta} < 0: No real solutions (two complex solutions)

• Practice with a quadratic equation:

  • Example: x^2 - 4x + 13

  • Calculate:

    • ext{delta} = (-4)^2 - 4(1)(13) = 16 - 52 = -36

    • Therefore, no real solutions.

Graphing Parabolas

• Identify the vertex and axis of symmetry:

  • x = - rac{b}{2a} gives the axis of symmetry.

• Substitute to find the vertex coordinates, indicating the maximum or minimum point of the parabola.

Systems of Linear Equations

• Using substitution or elimination methods.

• Example: For the equations, always solve for a particular variable to substitute into the other equation.

  • Solutions might yield infinite solutions or unique solutions depending on consistency of given equations.

Cubic Equations and Rational Root Theorem

• Rational Root Theorem helps find possible rational roots based on the divisors of the constant term.

• Example with cubic equation: Check divisors of a constant to verify possible roots.

• Practical solutions involve trying rational roots to identify correct equations easier.

Tips for Solving Quadratic Equations

• If polynomial division or substitution fails, revert to using the quadratic formula:

  • x = rac{-b ank{-} oot{d}}{2a} where:

    • $d = b^2 - 4ac$

  • Apply techniques based on the given problem's requirements.

Final Reminder:

• Thorough understanding and practicing multiple methods will help in recognizing patterns for problem-solving.