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: \( 2 + y = 5 \Rightarrow y = 3 \).