How to graph linear equations — slope-intercept, point-slope, two-point methods

Question

Graph the equation $2x + 3y = 12$ using two different methods. Also find its slope, x-intercept, and y-intercept.

(CBSE 9 & 10 board exam pattern)

Solution — Step by Step

The slope-intercept form is $y = mx + c$ , where $m$ is slope and $c$ is y-intercept.

2x + 3y = 12

3y = -2x + 12

y = -\frac{2}{3}x + 4

So slope $m = -2/3$ and y-intercept $c = 4$ (the point $(0, 4)$ ).

Set $y = 0$ : $2x + 0 = 12 \implies x = 6$

The x-intercept is $(6, 0)$ .

We have two points: $(0, 4)$ and $(6, 0)$ . Plot both on graph paper and draw a straight line through them. Two points are enough because a linear equation always gives a straight line.

Start at the y-intercept $(0, 4)$ . The slope $-2/3$ means: for every 3 units right, go 2 units down.

From $(0, 4)$ : move right 3 → $(3, ?)$ , down 2 → $(3, 2)$ . Plot this point. Draw the line through $(0, 4)$ and $(3, 2)$ .

Why This Works

A linear equation in two variables represents all $(x, y)$ pairs that satisfy it. These points form a straight line — that’s the geometric meaning of “linear.” The slope tells us the steepness and direction (positive = uphill, negative = downhill, zero = flat).

graph TD
    A["Graph a linear equation"] --> B{"Which form?"}
    B -->|"ax + by = c"| C["Find intercepts<br/>Set x=0, then y=0"]
    B -->|"y = mx + c"| D["Plot y-intercept<br/>Use slope to find next point"]
    B -->|"Two points given"| E["Plot both points<br/>Connect with straight line"]
    C --> F["Plot (0, c/b) and (c/a, 0)"]
    D --> G["From (0,c): rise=m, run=1"]
    F --> H["Draw line through points"]
    E --> H
    G --> H

Alternative Method — Table of Values

Make a table: pick any 3 values of $x$ , calculate $y$ , plot the points.

$x$	$y = (-2x + 12)/3$
0	4
3	2
6	0

Three points is a good check — if they’re not collinear, you’ve made an arithmetic mistake somewhere.

For CBSE boards: always plot at least 3 points (the marking scheme often requires it). Label the x-intercept, y-intercept, and slope on your graph. Use a scale that makes the graph fill at least half the graph paper provided.

Common Mistake

Students confuse slope with y-intercept. In $y = -\frac{2}{3}x + 4$ , the slope is $-2/3$ (the coefficient of $x$ ), NOT $4$ . The $4$ is the y-intercept. Also, a negative slope means the line goes downward from left to right — students sometimes draw it going upward and then wonder why the intercepts don’t match.

Question

Solution — Step by Step

Why This Works

Alternative Method — Table of Values

Common Mistake

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