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.
