System of Equations

A system of equations is a set of two or more equations involving the same set of variables. Solving these systems means finding the values of the variables that satisfy all equations simultaneously.

General Form:

\[ \begin{aligned} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{aligned} \] where: a and b are coefficients and c is a constant term independent of any variable

In engineering and applied sciences, systems of equations model real-world relationships such as force balance, circuit laws, material costs, and more.

Methods of Solving:

Substitution: Solve one equation for one variable and substitute it into the other.
Elimination (or Addition Method): Add or subtract equations to eliminate a variable, then solve the resulting equation.
Graphical Method: Graph each equation and identify the point(s) of intersection.
Matrix Method (for larger systems): Convert the system to matrix form and solve using row operations or inverse matrices.

Matrix Form:

For larger systems, the equations can be written in matrix form as:

\[ A\vec{x} = \vec{b} \]

Where:

$A$ is the coefficient matrix
$\vec{x}$ is the column vector of variables
$\vec{b}$ is the constant column vector

Solution Types:

Unique Solution: The system has one set of values for the variables (intersect at one point).
Infinitely Many Solutions: The equations represent the same line or plane.