Solving Systems of Equations: Substitution & Elimination Methods
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that share the same variables. The solution is the ordered pair $(x, y)$ that satisfies all equations simultaneously.
Standard Form of a System:
$$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} $$
The solution $(x, y)$ is the point of intersection of the two lines when graphed.
There are three possible outcomes for a system of two linear equations:
One Solution
Lines intersect at exactly one point. The system is consistent and independent.
No Solution
Lines are parallel (never intersect). The system is inconsistent.
Infinitely Many Solutions
Lines are identical (overlap). The system is consistent and dependent.
📊 Interactive Visualization: System of Equations
The graph below shows the system $ \begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases} $ with its solution at the intersection point.
Interactive visualization — available in the full lesson