What is a Graph?

A graph is a visual representation of numerical information. In mathematics, graphs help us see relationships between numbers or variables.

• Compare quantities.
  


• Understand trends.
  


• Solve equations and inequalities visually.
  

The Cartesian Plane

The Cartesian plane is a flat surface made of two perpendicular number lines:

• x-axis: horizontal line.
  


• y-axis: vertical line.
  

The point where the two axes meet is called the origin and is denoted by (0, 0).

Each point is identified by an ordered pair (x, y), where:

• x = horizontal distance from origin.
  


• y = vertical distance from origin.
  

Plotting Points on the Cartesian Plane

Steps to plot a point (x, y):

1. Start at the origin (0,0).
   
2. Move horizontally to x units (right if x>0, left if x < 0 ).


3. Move vertically to y units (up if y>0, down if y < 0 ).


4. Mark the point and label it.
   

Example: Plot (3,2)

Move 3 units right, 2 units up → mark the point.

Graph of a Linear Equation in Two Variables

A linear equation in two variables has the form: y = mx + c

• m = slope (steepness of the line)
  


• c = y-intercept (where line crosses y-axis)
  

Example: y = 2x + 1

Steps to draw the graph:

1. Choose at least two values for x and calculate y:

x	y = 2x + 1
0	1
1	3
2	5



2. Plot the points (0,1), (1,3), (2,5) on the Cartesian plane.


3. Draw a straight line through the points.


4. Label the line with the equation y = 2x + 1.

Cartesian Plane Example

y
6 |          
5 |         *
4 |          
3 |      *
2 |          
1 |   *
0 |---------------- x
   0  1  2  3  4

Points plotted: (0,1), (1,3), (2,5)

Key Points

• The graph of a linear equation is always a straight line.
  


• Positive slope → line rises; negative slope → line falls.
  


• x-intercept = value of x when y = 0; y-intercept = value of y when x = 0.
  


• At least two points are required to draw the line.
  

Practical Uses

• Represent relationships in real life (distance vs time, cost vs quantity).
  


• Solve problems graphically, including intersections of lines.
  


• Visualize inequalities when shading regions above or below a line.
  

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Graphs and Linear Equations in Two Variables

Real-Life Problems with Step-by-Step Solutions

1) Buying Notebooks

Problem: The cost of buying x notebooks is y = 150x + 200. Draw the graph for x = 0,1,2,3.

1. x = 0 → y = 200 → point (0,200)


2. x = 1 → y = 350 → point (1,350)


3. x = 2 → y = 500 → point (2,500)


4. x = 3 → y = 650 → point (3,650)


5. Plot points and draw a straight line.

y
650 |                   *
500 |             *
350 |       *
200 |  *
0   |_________________ x
     0  1  2  3

2) Cinema Tickets

Problem: Adult tickets = ₦500, children tickets = ₦300. Total revenue y for x adult tickets with 50 children tickets: y = 500x + 15000. Draw for x = 0,1,2,3.

1. x = 0 → y = 15000 → point (0,15000)


2. x = 1 → y = 15500 → point (1,15500)


3. x = 2 → y = 16000 → point (2,16000)


4. x = 3 → y = 16500 → point (3,16500)


5. Plot points and draw a line.

y
16500 |                      *
16000 |                 *
15500 |            *
15000 |      *
0     |_________________________ x
       0   1   2   3

3) Taxi Fare

Problem: Taxi charges ₦200 for first km, ₦100 each extra km. Total fare y = 200 + 100x. Draw for x = 0,1,2,3.

1. x = 0 → y = 200 → point (0,200)


2. x = 1 → y = 300 → point (1,300)


3. x = 2 → y = 400 → point (2,400)


4. x = 3 → y = 500 → point (3,500)


5. Plot points and draw line.

y
500 |                  *
400 |             *
300 |        *
200 |   *
0   |________________ x
     0 1 2 3

4) Selling Phones

Problem: Phone = ₦25000 each + ₦5000 shipping. Total cost y = 25000x + 5000. Draw for x = 0,1,2,3.

1. x = 0 → y = 5000 → point (0,5000)


2. x = 1 → y = 30000 → point (1,30000)


3. x = 2 → y = 55000 → point (2,55000)


4. x = 3 → y = 80000 → point (3,80000)


5. Plot points and draw line.

y
80000 |                       *
55000 |                  *
30000 |             *
5000  |        *
0     |________________ x
       0  1  2  3

5) Selling Eggs

Problem: Farmer sells x dozens of eggs at ₦600 per dozen + ₦200 packaging. Total revenue y = 600x + 200. Draw for x = 0,1,2,3.

1. x = 0 → y = 200 → point (0,200)


2. x = 1 → y = 800 → point (1,800)


3. x = 2 → y = 1400 → point (2,1400)


4. x = 3 → y = 2000 → point (3,2000)


5. Plot points and draw line.

y
2000 |                  *
1400 |             *
800  |        *
200  |   *
0    |________________ x
      0 1 2 3

6) Selling Water Bottles

Problem: Each bottle = ₦100. Fixed cost for delivery = ₦500. Total cost y = 100x + 500. Draw for x = 0,1,2,3.

1. x = 0 → y = 500 → point (0,500)


2. x = 1 → y = 600 → point (1,600)


3. x = 2 → y = 700 → point (2,700)


4. x = 3 → y = 800 → point (3,800)


5. Plot points and draw line.

y
800 |                  *
700 |             *
600 |        *
500 |   *
0   |________________ x
     0 1 2 3

7) Selling Books

Problem: Book = ₦200, delivery = ₦100. Total cost y = 200x + 100. Draw for x = 0,1,2,3.

1. x = 0 → y = 100 → point (0,100)


2. x = 1 → y = 300 → point (1,300)


3. x = 2 → y = 500 → point (2,500)


4. x = 3 → y = 700 → point (3,700)


5. Plot points and draw line.

y
700 |                  *
500 |             *
300 |        *
100 |   *
0   |________________ x
     0 1 2 3

8) School Transport

Problem: Bus fare = ₦150 per student. Extra charge = ₦50 for snacks. Total y = 150x + 50. Draw for x = 0,1,2,3.

1. x = 0 → y = 50 → point (0,50)


2. x = 1 → y = 200 → point (1,200)


3. x = 2 → y = 350 → point (2,350)


4. x = 3 → y = 500 → point (3,500)


5. Plot points and draw line.

y
500 |                  *
350 |             *
200 |        *
50  |   *
0   |________________ x
     0 1 2 3

9) Selling Fruit

Problem: Apples = ₦300 per kg, transport = ₦100. Total y = 300x + 100. Draw for x = 0,1,2,3.

1. x = 0 → y = 100 → point (0,100)


2. x = 1 → y = 400 → point (1,400)


3. x = 2 → y = 700 → point (2,700)


4. x = 3 → y = 1000 → point (3,1000)


5. Plot points and draw line.

y
1000 |                  *
700  |             *
400  |        *
100  |   *
0    |________________ x
      0 1 2 3

10) Printing Flyers

Problem: Printing cost = ₦50 per flyer + ₦500 fixed setup. Total y = 50x + 500. Draw for x = 0,1,2,3.

1. x = 0 → y = 500 → point (0,500)


2. x = 1 → y = 550 → point (1,550)


3. x = 2 → y = 600 → point (2,600)


4. x = 3 → y = 650 → point (3,650)


5. Plot points and draw line.

y
650 |                  *
600 |             *
550 |        *
500 |   *
0   |________________ x
     0 1 2 3