3.1 Systems of Linear Equations in Three Variables
1. Definition of Linear Equations in Three Variables
A linear equation in three variables is an equation that can be written in the standard form:
$$ax + by + cz = d$$
where $a, b, c,$ and $d$ are constants, and $a, b,$ and $c$ are not all zero. Geometrically, each linear equation in three variables represents a 2D plane in 3D space.
2. Types of Solutions in 3D Systems
A system of three linear equations in three variables can yield three distinct geometric outcomes:
- One Unique Solution: The three planes intersect at a single common point $(x, y, z)$.
- Infinite Solutions: The three planes intersect along a common straight line, or all three equations represent the exact same plane.
- No Solution: The planes are parallel or intersect in a way that there is no single point common to all three planes simultaneously.
3. Algebraic Solving Methods
- Elimination Method: Select two pairs of equations to eliminate one variable twice, reducing the system to two linear equations in two variables, then solve for the remaining variables.
- Substitution Method: Express one variable in terms of the other two from one equation, then substitute it into the other two equations to simplify to a two-variable system.
3.2 Simultaneous Equations Involving One Linear Equation and One Non-Linear Equation
1. Solving Simultaneous Equations (Linear & Non-Linear)
A non-linear equation contains terms with variables raised to powers other than 1 (e.g., $x^2, y^2, xy$).
Standard Solving Procedure:
- Rearrange the linear equation to express one variable as the subject (e.g., $y = mx + c$ or $x = g(y)$).
- Substitute this linear expression into the non-linear equation to eliminate that variable.
- Expand and simplify to form a single quadratic equation in one variable: $Ax^2 + Bx + C = 0$.
- Solve the quadratic equation using factorisation, completing the square, or the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$).
- Substitute the resulting values back into the linear subject equation to find the corresponding values of the second variable.
- Write solutions as ordered pairs $(x_1, y_1)$ and $(x_2, y_2)$.
2. Interpreting Points of Intersection and Discriminant
Geometrically, the solutions represent the intersection points between a straight line and a curve:
- $B^2 - 4AC > 0$: Line intersects the curve at 2 distinct points (2 real solutions).
- $B^2 - 4AC = 0$: Line is tangent to the curve, touching at 1 point (2 equal real solutions).
- $B^2 - 4AC < 0$: Line does not intersect the curve (no real solutions).
3. Applications and Problem Solving
When solving real-world word problems:
- Define variables clearly with appropriate units.
- Translate word constraints into algebraic equations (one linear, one non-linear).
- Solve the system and reject non-viable solutions (e.g., negative lengths, negative dimensions, or zero values where impossible).