Solving Systems of Linear Equations
The Substitution Method
Goal: Learn how to solve a system of two linear equations with two variables using the substitution method — an algebraic technique that gives exact answers every time.
1. What Is a System of Linear Equations?
A system of two linear equations with two variables is a pair of equations that must be true at the same time:
General Form:
$$\begin{cases} a_1\, x + b_1\, y = c_1 \\ a_2\, x + b_2\, y = c_2 \end{cases}$$where \(a_1, b_1, c_1, a_2, b_2, c_2\) are known constants and \(x, y\) are the unknowns.
A solution is an ordered pair $(x, y)$ that satisfies BOTH equations simultaneously.
Graphical meaning: Each linear equation represents a straight line on the coordinate plane. The solution of the system is the intersection point of the two lines.
Problem with graphing: Drawing lines by hand gives approximate answers. If the intersection is at $(2.3, -1.7)$, it's hard to read precisely from a graph. That's why we need algebraic methods!
Interactive Graph: System $x + y = 5$ and $x - y = 1$
The intersection point $(3, 2)$ is the solution. Drag the view to explore!
Interactive visualization — available in the full lesson