Contact Info

Location Engineering Building | room 332
Phone 435-797-1494
Contact Christian Bolander
Email christian.bolander@usu.edu

Hours

Mon–Thu	10am – 5pm
Fri	10am – 4pm

Drop-Ins Welcome!

Closed weekends and university holidays.

Systems of Linear Equations

Engineering Context

Systems of linear equations are abundant in the natural world around us. They represent instances when two or more variables or unknowns are used to describe how a system behaves. Because linear equations are easier to solve than their non-linear counterparts, engineers choose to work with systems of linear equations regularly by linearizing a non-linear problem. Thus the study and understanding of linear systems is critically important to an engineers’ work.

MAE: Reaction force N of a truck weighing W on a driveway inclined at θ with an applied force F

N \cos θ + F \sin θ = W

N \cos θ - F \cos θ = 0

ECE: A two-loop resistive circuit with three resistors R1, R2, and R3, two currents, I1 and I2, and two voltage sources V1 and V2

(R_{1} + R_{2}) I_{1} + R_{2} I_{2} = V_{1}

(R_{2} + R_{3}) I_{2} + R_{2} I_{1} = V_{2}

BE: Two-component blending of liquids of volume v1 and v2 (with combined volume V ) with concentrations c1 and c2 (combined to make C)

v_{1} + v_{2} = V

c_{1} v_{1} + c_{2} v_{2} = C V

CEE: External forces F1 and F2 acting on a truss supporting a load Tx horizontally and Ty vertically with an angle to a horizontal of θ

F_{1} \cos θ + F_{2} \cos θ - T_{x} = 0

F_{1} \cos θ - F_{2} \cos θ - T_{y} = 0

The Essentials

A system of linear equations or a linear system is a collection of two or more linear equations used to describe a system. A system of m equations with n variables can be written as

a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} ⋮ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} = b_{m}

Solutions to Linear Systems

[\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix}]

Solutions to Linear Systems

A solution to a linear system satisfies all of its equations simultaneously.
A linear system is consistent if it has at least one solution and inconsistent if it has none.
The solution to a linear system takes one of three forms: exactly one unique solution, no solution, or infinitely many solutions.

Systems of Linear Equations

Engineering Context

The Essentials

A Deeper Dive

Practice

More Resources