Introduction

A linear graph is a straight-line graph that shows the relationship between two quantities which change together at a constant rate. In real-life, many situations involve quantities that increase or decrease at a fixed rate — for example, cost with quantity, distance with time, or money earned with hours worked.

Quantitative reasoning helps us to interpret such real-life relationships using numbers, symbols, tables, and graphs.

Definition

A linear relationship between two variables x and y means that the value of y changes at a constant rate as x changes. It is represented by an equation of the form:

y = mx + c

where:

1. m is the slope or gradient (the rate of change of y with respect to x).
   


2. c is the y-intercept (the point where the line cuts the y-axis).
   


3. x and y are variables that depend on each other.
   

1. The cost of fuel depends on the number of litres bought.
   


2. The total amount of money spent depends on the number of items bought.
   


3. The distance travelled depends on the time taken if speed is constant.
   


4. The amount of pay depends on the number of hours worked.
   


5. The total cost of producing goods depends on the number of goods made.
   

1. Identify the two variables in the problem (usually x and y).
   


2. Write the equation that connects the variables.
   


3. Choose at least 3 values of x and calculate the corresponding values of y.
   


4. Make a table of values for x and y.
   


5. Plot the points (x, y) on the Cartesian plane.
   


6. Join the points with a straight line.
   


7. Label the graph and interpret the result.
   

Worked Examples

Example 1: Cost of Recharge Cards

A recharge card costs ₦100 each. Write an equation showing the relationship between the number of cards (x) and the total cost (y). Then draw the linear graph.

Solution:

Equation: y = 100x

Let x = 0, 1, 2, 3, 4

x (Number of cards)	y (Total cost ₦)
0	0
1	100
2	200
3	300
4	400

Plot the points: (0,0), (1,100), (2,200), (3,300), (4,400)

y
400 |                *
300 |           *
200 |      *
100 | *
0   |_________________ x
     0  1  2  3  4

Interpretation: The graph shows that the total cost increases in direct proportion to the number of recharge cards. The relationship is linear.

Example 2: Taxi Fare

A taxi driver charges ₦200 for starting the journey and ₦100 per kilometre travelled. Write the equation for the total fare (y) if the number of kilometres is x, and draw the graph.

Solution:

Equation: y = 100x + 200

Let x = 0, 1, 2, 3

x (km)	y (Fare ₦)
0	200
1	300
2	400
3	500

Plot the points: (0,200), (1,300), (2,400), (3,500)

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

Interpretation: The graph shows that as the distance increases, the fare increases at a constant rate of ₦100 per km.

Example 3: Quantitative Reasoning Application

A borehole pumping machine supplies 50 litres of water every minute. If the tank already has 100 litres of water, express the relationship between total volume (y) and time in minutes (x).

Solution:

Equation: y = 50x + 100

Let x = 0, 1, 2, 3, 4

x (minutes)	y (litres)
0	100
1	150
2	200
3	250
4	300

y
300 |               *
250 |          *
200 |     *
150 | *
100 |
0   |_________________ x
     0  1  2  3  4

Interpretation: The graph shows that the volume of water increases by 50 litres for every minute that passes.

1. The graph is always a straight line.
   


2. It can slope upwards (positive gradient) or downwards (negative gradient).
   


3. The y-intercept represents a fixed starting value.
   


4. When x increases by a constant amount, y also increases by a constant amount.
   

1. They help us to predict future values (e.g., cost, profit, distance).
   


2. They show how one quantity changes with another in a simple visual form.
   


3. They help in solving real-life problems involving constant rates (speed, wages, cost, etc.).
   


4. They are used in economics, science, and business planning.
   

Assignment / Practice Questions

1. A printing press charges ₦5000 setup fee plus ₦200 per poster. Write the equation connecting cost and number of posters, and draw the graph for 0–3 posters.
   


2. A car moves at a constant speed of 60 km/h. Draw the graph showing distance travelled for 0, 1, 2, 3 hours.
   


3. The pay of a factory worker is ₦1000 per hour plus a ₦2000 daily allowance. Write the equation and draw the graph for 0–4 hours.
   


4. When x = 0, y = 5. The rate of change of y with x is 3. Write the equation and draw the linear graph.
   


5. The total cost of buying oranges is ₦100 per dozen plus ₦50 transport fee. Draw the linear graph for 0–4 dozens.
   

Conclusion

Linear graphs are an important part of quantitative reasoning because they represent how two quantities relate in everyday life. By understanding how to form equations, make tables of values, and draw straight-line graphs, students can easily interpret and solve many real-world mathematical problems.