Linear Equations in Two Variables — Graphical Method Interpretation

Question

How do we solve a pair of linear equations in two variables graphically, and what do the different types of graphs tell us about the solution?

Solution — Step by Step

Each linear equation $ax + by + c = 0$ represents a straight line. To plot:

Find at least 2 points by substituting convenient values of $x$ (usually $x = 0$ and $y = 0$ )
Plot both lines on the same coordinate plane

For example, $x + y = 5$ and $x - y = 1$ :

Line 1: passes through $(0, 5)$ and $(5, 0)$
Line 2: passes through $(0, -1)$ and $(1, 0)$

The solution is the point where both lines meet — the coordinates $(x, y)$ of the intersection point satisfy BOTH equations simultaneously.

For our example, the lines meet at $(3, 2)$ , so $x = 3$ , $y = 2$ is the unique solution.

graph TD
    A[Two Linear Equations] --> B{How do the lines relate?}
    B -->|Intersect at one point| C[Unique Solution]
    C --> C1["Consistent, Independent"]
    C --> C2["a1/a2 is not equal to b1/b2"]
    B -->|Coincide, same line| D[Infinitely Many Solutions]
    D --> D1["Consistent, Dependent"]
    D --> D2["a1/a2 = b1/b2 = c1/c2"]
    B -->|Parallel, never meet| E[No Solution]
    E --> E1["Inconsistent"]
    E --> E2["a1/a2 = b1/b2 but not equal to c1/c2"]

For equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ :

Condition	Lines	Solution
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$	Intersecting	Unique
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Coincident	Infinite
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$	Parallel	None

This ratio test lets you predict the answer type before even drawing the graph — a huge time saver in exams.

Why This Works

A linear equation in two variables has infinitely many solutions — each one is a point on the line. When we have TWO equations, we need a point that lies on BOTH lines. Geometrically, that is the intersection point. If the lines never meet (parallel), no such point exists. If they overlap completely, every point on the line works.

In CBSE Class 10 boards, the graphical method question typically carries 4-5 marks. You will be asked to draw the graph AND state whether the system is consistent or inconsistent. Always label your axes, mark the intersection point clearly, and write the solution as an ordered pair.

Alternative Method

Instead of graphing, use algebraic methods (substitution or elimination) to find the exact solution. Graphing gives approximate answers limited by your drawing accuracy, while algebra gives exact values. For exam accuracy, use algebra and verify with the ratio test.

Common Mistake

Students often confuse “coincident lines” with “intersecting lines.” Coincident means the two equations are actually the same line (like $x + y = 3$ and $2x + 2y = 6$ ). They do NOT intersect at a single point — they overlap everywhere. If you misread this as “one solution,” you lose marks. Check the ratio: if all three ratios $a_1/a_2 = b_1/b_2 = c_1/c_2$ are equal, the lines coincide.