Understanding Linear Systems of Equations: Definitions, Solutions, and Consistency

Introduction to Linear Equations

A linear equation is a fundamental concept in mathematics used to describe relationships between variables that form a straight line. Its general form can be written as:

mx q 0

This form indicates that the highest power of the variable is 1. Examples of linear equations include:

x - 2 8 3x - 5 0

These equations generate a straight line when their solutions are plotted on a coordinate plane.

Systems of Linear Equations

A system of linear equations involves multiple equations that share the same set of variables. The purpose of such a system is to find the points where the lines (or planes, in higher dimensions) intersect, which are the simultaneous solutions of the equations.

For example, consider the following system with one variable:

3x - 15 0 2x -18

The solutions to the first equation are:

x 5

The solutions to the second equation are:

x -9

Since these solutions do not coincide, the system has no solution and the intersection is empty.

In a system of linear equations with more than one variable, the solutions must satisfy all the equations simultaneously. For instance, consider the system:

3x y - 1 0

y 10

This example can be solved using the substitution method. By substituting y 10 into the first equation, we get:

3x 10 - 1 0

Which simplifies to:

3x 9 0

And solving for x yields:

x -3

Substituting this value back into the second equation:

y 10

We find the solution to be:

x -3, y 10

Consistency and Determinants

In the context of systems of linear equations, a system is consistent if it has at least one solution, and inconsistent if it has no solutions. This consistency can be determined by the determinant of the coefficient matrix A in the matrix form AX B, where:

A is the coefficient matrix. X is the variable vector. B is the constant vector.

If the determinant of A is non-zero, the system is consistent and has a unique solution. If the determinant of A is zero, the system is inconsistent or may have infinitely many solutions.

Example Solution Using Substitution

Consider this system:

2y x - 9 … equation 1

x y - 6 … equation 2

To solve for x and y using substitution, follow these steps:

Select an equation. Equation 2 is already in the form x y - 6, so we substitute this into equation 1. Substitute x in equation 1:2y (y - 6) - 9 Simplify:2y y - 15 Rearrange to solve for y:- y -15 Solve for y:y 15 Substitute y back into equation 2:x 15 - 6 Solve for x:x 9

The solution to this system is hence ( text{x} 9 ) and ( text{y} 15 ).