Unit 3: Equations

Simultaneous Linear Equations in Two Variables

A system of simultaneous linear equations is a set of two or more linear equations with the same variables. We look for a pair (x, y) that satisfies all equations at the same time.

The standard form is:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Methods for Solving

1. Substitution Method:
   1. Solve one equation for one variable (e.g., solve for y in terms of x).
   2. Substitute this expression into the other equation.
   3. This creates a new equation with only one variable (x). Solve it.
   4. Substitute the x-value back into your expression from step 1 to find y.
2. Elimination Method:
   1. Multiply one or both equations by constants so that the coefficients of one variable (e.g., y) are opposites (like -3y and +3y).
   2. Add the two new equations together. This will eliminate one variable.
   3. Solve the resulting equation for the remaining variable.
   4. Substitute this value back into one of the original equations to find the other variable.

Nature of Solutions (Graphical Interpretation)

The nature of the solution depends on the relationship between the coefficients.

• Unique Solution: If a₁/a₂ ≠ b₁/b₂

  The lines intersect at exactly one point. This is the most common case.

• No Solution: If a₁/a₂ = b₁/b₂ ≠ c₁/c₂

  The lines are parallel and never intersect.

• Infinitely Many Solutions: If a₁/a₂ = b₁/b₂ = c₁/c₂

  The lines are coincident (they are the same line).

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The standard form is:

ax² + bx + c = 0 (where a, b, and c are constants, and a ≠ 0)

Methods of Solution

1. Factoring: If the quadratic expression can be factored, set each factor to zero.
   • Example: x² - 5x + 6 = 0
     (x - 2)(x - 3) = 0
     So, x = 2 or x = 3.
2. Quadratic Formula: This formula solves any quadratic equation.

   x = [ -b ± √(b² - 4ac) ] / (2a)

Nature of Roots (Using the Discriminant)

The Discriminant (D) is the part under the square root in the quadratic formula:

D = b² - 4ac

The value of D tells us about the type of roots without having to solve the equation:

• If D > 0: The equation has two distinct real roots.
• If D = 0: The equation has one real root (or two equal real roots).
• If D < 0: The equation has no real roots (it has two complex conjugate roots).

Sum and Product of Roots

For a quadratic equation `ax² + bx + c = 0`, if the roots are α (alpha) and β (beta), we have a direct relationship:

Sum of Roots: α + β = -b / a
Product of Roots: αβ = c / a