Simultaneous Linear Equations

Introduction

Simultaneous linear equations involve finding the values of variables that satisfy two or more linear equations. In VCE Mathematical Methods, we focus on systems with two or three variables. Understanding the nature of solutions (unique, infinite, or none) is crucial.

Solving Systems of Two Linear Equations

Methods

1. Substitution: Solve one equation for one variable and substitute that expression into the other equation.
2. Elimination: Multiply one or both equations by constants so that the coefficients of one variable are opposites. Add the equations to eliminate that variable.

Geometric Interpretation

Each linear equation in two variables represents a straight line on the Cartesian plane. The solution to the system represents the point(s) where the lines intersect.

• Unique Solution: The lines intersect at one point. The system is consistent and independent.
• Infinite Solutions: The lines are coincident (the same line). The system is consistent and dependent.
• No Solution: The lines are parallel and distinct. The system is inconsistent.

Examples

Example 1: Unique Solution

Solve the system:

$$x + y = 5$$
$$x - y = 1$$

Adding the two equations:

$$2x = 6 \Rightarrow x = 3$$

Substituting $x = 3$ into the first equation:

$\$3 + y = 5 \Rightarrow y = 2$$

Solution: $(3, 2)$

Example 2: Infinite Solutions

Solve the system:

$$x + y = 3$$
$$2x + 2y = 6$$

Notice that the second equation is simply twice the first equation. These represent the same line. There are infinite solutions. We can express the solutions as $y = 3 - x$, where $x$ can be any real number.

Example 3: No Solution

Solve the system:

$$x + y = 2$$
$$x + y = 5$$

Subtracting the first equation from the second:

$\$0 = 3$$

This is a contradiction, indicating no solution. The lines are parallel.

Solving Systems of Three Linear Equations

Methods

1. Elimination: Use elimination to reduce the system to two equations in two variables. Then solve the resulting system.
2. Matrix Methods: (Beyond the scope of this key knowledge point, but relevant in later topics).

Nature of Solutions

In three dimensions, each linear equation represents a plane. The solution to the system represents the intersection of these planes.

• Unique Solution: The planes intersect at a single point.
• Infinite Solutions: The planes intersect in a line or are coincident (the same plane).
• No Solution: The planes do not have a common intersection (e.g., parallel planes, planes intersecting pairwise but not at a common point).

Example

Solve the system:

$$x + y + z = 6$$
$$2x - y + z = 3$$
$$x + 2y - z = 2$$

Add the first and third equations:

$$2x + 3y = 8$$

Subtract the second equation from the first:

$$-x + 2y = 3$$

Multiply the second equation by 2:

$$-2x + 4y = 6$$

Add this to the equation $2x + 3y = 8$:

$$7y = 14 \Rightarrow y = 2$$

Substitute $y = 2$ into $-x + 2y = 3$:

$$-x + 4 = 3 \Rightarrow x = 1$$

Substitute $x = 1$ and $y = 2$ into $x + y + z = 6$:

$\$1 + 2 + z = 6 \Rightarrow z = 3$$

Solution: $(1, 2, 3)$

Cases of No Solution or Infinite Solutions

Identifying these cases often involves observing inconsistencies or dependencies in the equations. Technology (CAS calculator) can be helpful, but it’s important to understand the underlying concepts.

Key Considerations

• Parallel Planes/Lines: If equations represent parallel planes or lines, there is no solution.
• Dependent Equations: If one equation is a multiple of another, there are infinite solutions.
• Inconsistent Equations: If manipulations lead to a contradiction (e.g., \$0 = 1$), there is no solution.

Summary

Understanding the geometric interpretations and the algebraic techniques for solving simultaneous linear equations is crucial for VCE Mathematical Methods. Be prepared to identify the nature of the solutions and to solve systems using appropriate methods.