MathsICSE Class 9 · Coordinate Geometry

Graphs of Linear Equations and Graphical Solution of Simultaneous Equations (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

The graph of a linear equation ax + by = c is a straight line, and every point on it is a solution. To solve two such equations graphically, draw both on the same axes and scale; the intersection point gives the solution. Parallel lines give no solution and coinciding lines give infinitely many.

The two lines cross at (3, 2), the only point on both, so it is the solution; parallel lines never meet and coinciding lines share every point.

Illustration drawn for QuestionBuddy.

Understand Graphs of Linear Equations and Solving Simultaneous Equations Graphically

Variables. In $y = 2 x + 3$ you choose $x$ (the independent variable) and calculate $y$ (the dependent variable). The independent variable goes on the x-axis.

Quadrants and plotting. The axes divide the plane into four quadrants: I $(+, +)$ , II $(-, +)$ , III $(-, -)$ , IV $(+, -)$ . A point on the x-axis has $y = 0$ ; on the y-axis $x = 0$ . The order matters: $(2, 5) \neq = (5, 2)$ .

Drawing a line. Make a table with at least three values of $x$ , find $y$ , plot the points, and join them with a ruler. Two points fix the line; the third is a check, and if the three are not collinear there is an arithmetic slip. The intercepts are quick: put $x = 0$ to get the y-intercept, and $y = 0$ for the x-intercept. Always state the scale (for example 1 cm = 1 unit) and label the axes.

Reading a point from a line. To check that $(p, q)$ lies on a line, substitute it into the equation. To find an unknown coordinate, substitute the known one and solve.

Graphical solution. Draw both lines on the same axes with the same scale. The point of intersection satisfies both equations.

Intersecting lines: one solution, the coordinates of the point.
Parallel lines (equal slopes, different intercepts): no solution.
Coinciding lines: infinitely many solutions.

For $a_{1} x + b_{1} y = c_{1}$ and $a_{2} x + b_{2} y = c_{2}$ : one solution if $\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$ ; no solution if $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$ ; infinitely many if all three ratios are equal.

Areas from graphs. Lines and axes often enclose a triangle. Read the vertices off the graph (or from the intercepts) and use $\frac{1}{2} \times base \times height$ .

Level. Class IX is a school-exam class. Exam questions are mostly 3 to 5 mark graph questions; the area-of-triangle and parameter-for-no-solution items here are at the top of that range and labelled honestly.

Graphs of Linear Equations and Solving Simultaneous Equations Graphically: formulas to remember

Intercepts: $x = 0 \to y -intercept; y = 0 \to x -intercept$; Quick way to fix a line.
Point on a line: $(p, q) \in a x + b y = c ⟺ a p + b q = c$; Substitute to check.
One solution: $\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$; Lines intersect.
No solution: $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$; Parallel lines.
Area of a triangle: $\frac{1}{2} \times base \times height$; Vertices read from the graph.

How to solve Graphs of Linear Equations and Solving Simultaneous Equations Graphically questions

Table, plot, join, read, check.

1Make a table of three or more values from the equation.
2Choose a scale that fits all the points; state it.
3Plot the points and draw the line with a ruler, extending it.
4For two equations, draw both on the same axes and read the intersection.
5Check the point in both equations.
6Write the solution as $x = \dots, y = \dots$ .

Graphs of Linear Equations and Solving Simultaneous Equations Graphically in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Two taxi fares

Suppose Taxi A charges ₹20 plus ₹10 per km, so the fare is $y = 10 x + 20$ , and Taxi B charges ₹40 plus ₹5 per km, so $y = 5 x + 40$ , where $x$ is the distance in km.

At what distance are the fares equal, and which taxi is cheaper for a 10 km trip?

Try it first, then show the solutionSolution

Equal when $10 x + 20 = 5 x + 40$ , so $x = 4$ km, fare ₹60 (the intersection point $(4, 60)$ ).
At 10 km: A charges ₹120, B charges ₹90.

Answer: Equal at 4 km (₹60); Taxi B is cheaper for 10 km (₹90 against ₹120).

Memory hook: The intersection is where both equations agree.

Reading a straight-line cost

Imagine a printing shop charging $y = 2 x + 10$ rupees for $x$ pages, where 10 is a fixed set-up fee.

What are the intercepts and what do they mean?

Try it first, then show the solutionSolution

At $x = 0$ : $y = 10$ , the set-up fee.
At $y = 0$ : $x = - 5$ , which has no meaning as pages, so only the part with $x \geq 0$ is used.

Answer: y-intercept 10 (the fixed fee); x-intercept −5 (not meaningful here).

Memory hook: The y-intercept is the cost for zero units.

Tricks to remember Graphs of Linear Equations and Solving Simultaneous Equations Graphically

Short memory aids. Say each one aloud once, then use it on a question the same day.

Three-point rule
Always take three points; if they are not in a line, recheck.
Use it when: Plotting accurately.
Cross at the answer
Where lines cross is where both equations are true.
Use it when: Graphical solution.
Ratios decide
Compare a1/a2, b1/b2 and c1/c2 to know the type of solution.
Use it when: No-solution and infinite-solution questions.

Graphs of Linear Equations and Solving Simultaneous Equations Graphically: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Quadrants · 1 mark · MCQBoard level

The point $(- 3, 4)$ lies in:
(a) Quadrant I
(b) Quadrant II
(c) Quadrant III
(d) Quadrant IV

Try it first, then show the solutionSolution

The abscissa is negative and the ordinate is positive.

Answer: (b) Quadrant II

Example 2: Table and intercepts · 3 marks · Short answerBoard level

Draw the table for $y = 2 x - 1$ for $x = - 1, 0, 1, 2, 3$ and find where the line crosses the two axes.

Try it first, then show the solutionSolution

$y = - 3, - 1, 1, 3, 5$ respectively.
At $x = 0$ : $y = - 1$ , so it meets the y-axis at $(0, - 1)$ .
At $y = 0$ : $x = \frac{1}{2}$ , so it meets the x-axis at $(\frac{1}{2}, 0)$ .

Answer: Points $(- 1, - 3), (0, - 1), (1, 1), (2, 3), (3, 5)$ ; axes at $(0, - 1)$ and $(\frac{1}{2}, 0)$ .

Example 3: Graphical solution and area · 5 marks · Long answerHarder board level

Solve graphically: $x + y = 7$ and $x - y = 1$ . Find the area of the triangle bounded by the two lines and the x-axis.

Try it first, then show the solutionSolution

$x + y = 7$ passes through $(0, 7), (3, 4), (7, 0)$ ; $x - y = 1$ passes through $(1, 0), (4, 3), (0, - 1)$ .
They meet at $(4, 3)$ : check $4 + 3 = 7$ and $4 - 3 = 1$ . So $x = 4$ , $y = 3$ .
The lines meet the x-axis at $(7, 0)$ and $(1, 0)$ : base $= 6$ , height $= 3$ (the y-coordinate of the intersection).
Area $= \frac{1}{2} \times 6 \times 3 = 9$ square units.

Answer: $x = 4$ , $y = 3$ ; area $9$ square units.

Example 4: Intercepts and an unknown coordinate · 4 marks · Short answerHarder board level

For the line $3 x - 2 y = 12$ , find the intercepts, the area of the triangle formed with the axes, and the abscissa of the point with ordinate $3$ .

Try it first, then show the solutionSolution

$y = 0$ : $x = 4$ , so $(4, 0)$ . $x = 0$ : $y = - 6$ , so $(0, - 6)$ .
Area $= \frac{1}{2} \times 4 \times 6 = 12$ square units.
$y = 3$ : $3 x - 6 = 12$ , $x = 6$ .

Answer: Intercepts $4$ and $- 6$ ; area $12$ square units; abscissa $6$ .

Example 5: Kinds of solution · 4 marks · Short answerHOTS / competency

(i) How many solutions do $x + 2 y = 4$ and $2 x + 4 y = 8$ have? (ii) What about $x + 2 y = 4$ and $2 x + 4 y = 9$ ? (iii) Find $k$ for which $2 x + 3 y = 6$ and $4 x + k y = 5$ have no solution.

Try it first, then show the solutionSolution

(i) Ratios $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$ : the lines coincide, so infinitely many solutions.
(ii) $\frac{1}{2} = \frac{2}{4} \neq = \frac{4}{9}$ : parallel, no solution.
(iii) Need $\frac{2}{4} = \frac{3}{k} \neq = \frac{6}{5}$ . From $\frac{1}{2} = \frac{3}{k}$ , $k = 6$ ; then $\frac{c _{1}}{c _{2}} = \frac{6}{5} \neq = \frac{1}{2}$ , so no solution.

Answer: (i) infinitely many (ii) none (iii) k = 6

Where students lose marks in Graphs of Linear Equations and Solving Simultaneous Equations Graphically

The slips examiners see most often, and what to write instead.

Mistake: Reversing the coordinates
The abscissa comes first.
Wrong working
Plotting $(2, 5)$ as 5 right and 2 up.
Right working
2 right and 5 up.
Do this: x then y, always.
Usually costs: Plot marks.
Mistake: Drawing the line through only two points
An arithmetic slip goes unnoticed.
Wrong working
Two points, no check.
Right working
Three points; the third confirms the line.
Do this: Table with three rows.
Usually costs: Accuracy marks.
Mistake: Reading the solution off two different scales
The intersection depends on one common scale.
Wrong working
Different scales on the two lines.
Right working
One pair of axes and one scale for both lines.
Do this: Draw both on the same graph.
Usually costs: Final answer.

Exam tips

Class IX is examined in school papers; graphs are drawn on graph paper with a stated scale.
Choose the scale so that the intersection point is on the sheet before you plot; take about 5 values of x.
Always verify the intersection by substituting into both equations.
For "area of triangle" questions, mark the three vertices clearly first.

Practice questions on Graphs of Linear Equations and Solving Simultaneous Equations Graphically

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The point $(- 4, 0)$ lies on:
(a) the positive x-axis
(b) the negative x-axis
(c) the positive y-axis
(d) the negative y-axis
Hint
$y = 0$ means on the x-axis.
Answer
(b)
Q2 · 3 marks · Short answerHarder board level
For the line $2 x - 3 y = 6$ , find the intercepts. Does $(6, 2)$ lie on it? What is the abscissa when the ordinate is $4$ ?
Hint
Substitute.
Answer
Intercepts $(3, 0)$ and $(0, - 2)$ ; $12 - 6 = 6$ , so yes; $2 x - 12 = 6$ gives $x = 9$ .
Q3 · 5 marks · Long answerHarder board level
Solve graphically $x + 2 y = 8$ and $x - y = 2$ . Find the area of the triangle formed by the two lines and the y-axis.
Hint
They meet at (4, 2).
Answer
$x = 4$ , $y = 2$ (check $4 + 4 = 8$ , $4 - 2 = 2$ ). The lines meet the y-axis at $(0, 4)$ and $(0, - 2)$ : base $6$ , height $4$ (the x-coordinate), area $\frac{1}{2} \times 6 \times 4 = 12$ square units.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The pair $x + 2 y = 3$ and $2 x + 4 y = 7$ has no solution.
Reason (R): The lines are parallel, so they never intersect.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$\frac{1}{2} = \frac{2}{4} \neq = \frac{3}{7}$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: Taxi A charges ₹20 plus ₹10 per km ( $y = 10 x + 20$ ); Taxi B charges ₹40 plus ₹5 per km ( $y = 5 x + 40$ ).
(a) Which is the independent variable?
(b) Find the fare of Taxi A for 6 km.
(c) At what distance are the fares equal and what is that fare?
(d) Which taxi is cheaper for 10 km?
Hint
Set 10x + 20 = 5x + 40.
Answer
(a) distance x (b) ₹80 (c) 4 km, ₹60 (d) Taxi B: ₹90 against ₹120.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the abscissa?Show answer
The x-coordinate of a point.
2.How many points fix a line?Show answer
Two (a third checks it).
3.What do parallel lines mean for a pair of equations?Show answer
There is no solution.
4.Which quadrant has both coordinates negative?Show answer
Quadrant III.

Graphs of Linear Equations and Solving Simultaneous Equations Graphically: frequently asked questions

Do I need graph paper?

The syllabus asks for graphical solution, so yes: use graph paper with a stated scale.

What if the lines meet at a non-integer point?

Read it as accurately as the scale allows, then verify algebraically.

What does the intercept mean in a real problem?

The value of the dependent variable when the independent variable is zero.