Linear graphs and their applications: Class 9 Mathematics Notes, Short Questions and Numericals

The Cartesian plane and plotting points, graphs of linear equations, slope and intercepts, solving two simultaneous equations by graph, and applications such as distance and time, conversion and cost graphs.

Key points

The Cartesian plane is formed by two number lines that cross at right angles at the origin O. The horizontal line is the x-axis and the vertical line is the y-axis. They divide the plane into four quadrants. A point is written as an ordered pair (x, y): x is the distance along the x-axis and y is the distance along the y-axis. The origin is (0, 0).

Signs of coordinates in the four quadrants:

Quadrant I: x positive and y positive, written (+, +).
Quadrant II: (−, +).
Quadrant III: (−, −).
Quadrant IV: (+, −).
Points on the x-axis have y = 0 and points on the y-axis have x = 0.

To plot a point (a, b), start at the origin, move a units along the x-axis (right if positive, left if negative), then b units parallel to the y-axis (up if positive, down if negative). Mark the point and label it. A suitable scale must be chosen on each axis, such as 1 cm for 1 unit, and written on the graph.

A linear equation in two variables has the form ax + by + c = 0, where a and b are not both zero. It has infinitely many solutions, each an ordered pair (x, y). When these pairs are plotted, they all lie on one straight line, which is the graph of the equation. A straight line is fixed by two points, but a third point is used as a check.

To draw the graph, make a table. Choose three values of x, find the matching y from the equation, plot the points and join them with a ruler. For y = 2x + 1: x = 0 gives y = 1, x = 1 gives y = 3, and x = 2 gives y = 5. The points (0, 1), (1, 3) and (2, 5) lie on one line. Extend the line with arrows at both ends.

Special lines:

x = a is a vertical line, parallel to the y-axis, through (a, 0).
y = b is a horizontal line, parallel to the x-axis, through (0, b).
y = mx passes through the origin.
x = 0 is the y-axis itself and y = 0 is the x-axis.

The slope (gradient) m of a line measures how steep it is: m = rise / run = (y₂ − y₁) / (x₂ − x₁) for two points (x₁, y₁) and (x₂, y₂) on the line. A positive slope means the line rises as x increases, and a negative slope means it falls. A horizontal line has slope 0. A vertical line has no defined slope.

In the form y = mx + c, m is the slope and c is the y-intercept, the value of y where the line cuts the y-axis. The x-intercept is found by putting y = 0. Intercepts give the quickest way to draw a line: join (0, c) and the x-intercept.

Two equations in two unknowns form a pair of simultaneous equations. Draw both lines on the same axes. The point where they cross satisfies both equations, so its coordinates are the solution. If the lines are parallel there is no solution. If they coincide, there are infinitely many. Always check the answer by substituting it into both equations.

Graphs are used to show real situations. In a distance-time graph, time is on the x-axis and distance on the y-axis. The slope of the line is the speed. A straight line means constant speed, and a horizontal part means the object is at rest. Speed = distance ÷ time, in units such as km/h or m/s.

A conversion graph changes one unit into another, such as degrees Celsius and Fahrenheit using F = (9/5)C + 32, or one currency into another at a fixed rate. Pick values on one axis and read the matching values on the other. A cost graph, for example a taxi fare y = 20x + 50 where x is distance in km, has the fixed charge as intercept and the rate per km as slope.

To read values from a graph, draw a line from the given value on one axis to the graph, then straight to the other axis, and read off the scale. Estimates from graphs depend on the accuracy of drawing, so use a sharp pencil and a ruler.

Study tip: Use graph paper, a sharp pencil and a ruler, and write the scale, the axis labels and the equation of each line on the graph. Prepare the table of values neatly, because marks are given for it. For application questions, state what the slope and intercept mean in words, with units.

Definitions and flashcards

Say the meaning first, then open the card to check.

Origin

The point (0, 0) where the axes meet.

Ordered pair

Two numbers (x, y) giving the position of a point.

Quadrant

One of the four regions into which the axes divide the plane.

Slope

The ratio of vertical change to horizontal change between two points of a line.

y-intercept

The y-coordinate of the point where a line cuts the y-axis.

Simultaneous equations

Two or more equations satisfied by the same values of the unknowns.

Scale

The number of units each division of an axis represents.

Conversion graph

A straight line graph for changing from one unit to another.

Formula sheet

Name	Formula	What it means
Slope	m = (y₂ − y₁) / (x₂ − x₁)	Steepness of the line through two points.
Slope-intercept form	y = mx + c	m is the slope and c the y-intercept.
Speed	speed = distance / time	Slope of a distance-time graph.
Temperature conversion	F = (9/5)C + 32	Celsius to Fahrenheit.
Midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2)	Point halfway between two points.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

What are the coordinates of the origin? In which quadrant does (−3, 4) lie? (2 marks)

The origin is (0, 0). The point (−3, 4) has a negative x and a positive y, so it lies in the second quadrant.

What is the graph of x = 4? (2 marks)

The graph of x = 4 is a vertical straight line parallel to the y-axis. Every point on it has x-coordinate 4, for example (4, −2), (4, 0) and (4, 5).

What does the slope of a distance-time graph represent? (2 marks)

The slope is the change in distance divided by the change in time, which is the speed. A steeper line means a greater speed, and a horizontal line means the object is not moving.

Find the slope of the line through (1, 2) and (4, 8). (3 marks)

Slope m = (y₂ − y₁) / (x₂ − x₁) = (8 − 2) / (4 − 1) = 6/3 = 2. The positive slope shows that the line rises from left to right.

Explain how to solve two simultaneous linear equations by the graphical method. (3 marks)

Make a table of values for each equation and plot both lines on the same axes with the same scale. The point where the two lines cross lies on both lines, so its coordinates satisfy both equations. Read the coordinates of this point; that is the solution. Finally substitute the values in both equations to check them.

A taxi charges Rs 50 plus Rs 20 for each km. Write the cost equation, state its slope and intercept, and find the cost of a 6 km journey. (5 marks)

Let x be the distance in km and y the cost in rupees. Then y = 20x + 50. This is of the form y = mx + c, so the slope is 20 (rupees per km) and the y-intercept is 50 (the fixed charge). For x = 6, y = 20 × 6 + 50 = 120 + 50 = 170. The cost is Rs 170. The graph is a straight line through (0, 50) and (6, 170).

Numericals with worked steps

Prepare a table for y = 2x + 1 for x = 0, 1, 2 and state the slope and y-intercept.

x = 0: y = 2(0) + 1 = 1.
x = 1: y = 2(1) + 1 = 3.
x = 2: y = 2(2) + 1 = 5.
Compare with y = mx + c: m = 2 and c = 1.

Answer: Points (0, 1), (1, 3), (2, 5); slope 2 and y-intercept 1

Solve y = x + 1 and y = −x + 5 by finding where the lines meet.

At the meeting point both y values are equal: x + 1 = −x + 5.
2x = 4, so x = 2.
y = 2 + 1 = 3.
Check in the second equation: −2 + 5 = 3.

Answer: (2, 3)

Convert 40 degrees Celsius to Fahrenheit using F = (9/5)C + 32.

F = (9/5) × 40 + 32.
(9/5) × 40 = 72.
F = 72 + 32 = 104.

Answer: 104 °F

A car travels 150 km in 2.5 hours at constant speed. Find the speed and the distance covered in 4 hours.

Speed = distance / time = 150 / 2.5 = 60 km/h.
Distance in 4 h = 60 × 4 = 240 km.
The graph is a straight line from (0, 0) with slope 60.

Answer: 60 km/h and 240 km

Find the slope of the line through (−2, 3) and (4, −9).

m = (y₂ − y₁)/(x₂ − x₁) = (−9 − 3)/(4 − (−2)).
= −12 / 6.
= −2.

Answer: Slope = −2 (the line falls as x increases)

Common mistakes

Writing the coordinates in the wrong order. The point (3, 5) is 3 along and 5 up, not the reverse.
Choosing a scale that is not uniform or leaving the scale unmarked on the axes.
Plotting only two points, so that an error in a calculation goes unnoticed. Use a third check point.
Subtracting in a different order for the numerator and the denominator when finding slope.
Reading the solution of simultaneous equations from the graph without checking it in both equations.

Practise this chapter

Practice questions and timed test Printable worksheet