Linear Algebra
Study of vectors, matrices, and linear transformations.
Basic Concepts
Systems of Linear Equations
What Are Systems of Linear Equations?
A system of linear equations is a set of equations where each one is a straight line. You're often asked to find where all the lines cross, which is the solution to the system.
\[ \begin{align*} x + y &= 5 \ 2x - y &= 1 \end{align*} \]
Solving Systems
- Graphing: Draw each line and see where they meet.
- Substitution: Solve one equation for one variable, then plug it into the other.
- Elimination: Add or subtract equations to eliminate a variable.
- Matrix Methods: Use matrices to solve big systems quickly.
Why Is This Important?
Solving systems helps us:
- Predict how things interact, like supply and demand.
- Balance chemical equations.
- Design circuits and networks.
Example
For the system above:
- Add the two equations to eliminate \( y \): \( (x + y) + (2x - y) = 5 + 1 \Rightarrow 3x = 6 \Rightarrow x = 2 \).
- Substitute \( x \) into the first equation: \( 2 + y = 5 \Rightarrow y = 3 \).
Examples
Solving two equations in two unknowns to find where lines cross.
Finding values for variables that satisfy multiple relationships.
In a Nutshell
Systems of linear equations help us find where lines (or planes) intersect.