Graphical Method

In the previous lesson we saw that one linear equation in two variables has infinitely many solutions. What happens when we require two equations to hold at the same time?

Systems of Equations

📐Definition — System of Linear Equations

A system of linear equations (or linear system) is a set of two or more linear equations involving the same variables. A solution of the system is an ordered pair $(x, y)$ that satisfies every equation in the system simultaneously.

For a system of two linear equations in two unknowns, the graphical approach is intuitive: graph both lines and look at where they meet.

The Graphical Method

Each equation defines a line. The solution to the system is any point that lies on both lines simultaneously — that is, their point of intersection.

⚡Theorem — Three Cases for Two Lines

A system of two linear equations in two variables has exactly one of three outcomes:

One solution (consistent, independent): the lines intersect at exactly one point.
No solution (inconsistent): the lines are parallel (same slope, different intercepts) and never meet.
Infinitely many solutions (consistent, dependent): the lines are identical, so every point on one line is also on the other.

Interactive Explorer

Use the explorer below to see how the solution changes as you adjust the equations. Try making the lines parallel, then identical, to observe all three cases.

Line 1: 1x + 1y = 3

a₁=b₁=c₁=

Line 2: 2x - 1y = 1

a₂=b₂=c₂=

Solution: x = 1.333, y = 1.667

Worked Example

✎Example — Solving a System Graphically

Solve graphically:

$\begin{cases} x + y = 4 \\ x - y = 2 \end{cases}$

Solution. Graph each line by finding two points.

Line 1 ( $x + y = 4$ ):

When $x = 0$ : $y = 4$ — point $(0, 4)$
When $y = 0$ : $x = 4$ — point $(4, 0)$

Line 2 ( $x - y = 2$ ):

When $x = 0$ : $y = -2$ — point $(0, -2)$
When $y = 0$ : $x = 2$ — point $(2, 0)$

The two lines intersect at $(3, 1)$ .

Verification: $3 + 1 = 4$ ✓ and $3 - 1 = 2$ ✓.

Precision and Limitations

ℹNote — When Graphical Methods Fall Short

The graphical method is excellent for visualization and understanding the geometric meaning of a system. However, reading coordinates from a graph is inherently approximate. If the solution involves fractions like $\left(\dfrac{16}{7}, \dfrac{3}{7}\right)$ , a graph alone will not give exact values. For exact solutions, use the algebraic methods covered in the next lesson.

The graphical method is a powerful first step: it tells you how many solutions to expect and gives an approximate answer. Algebraic methods then provide the precision.