EconomicsISC Class 11 · Mathematical Tools Used in Economics

Variables, Functions, Graphs and the Straight Line (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

A variable is a quantity that can take different values; in a function y = f(x), y (dependent variable) depends on x (independent variable). A straight line is y = a + bx, where a is the intercept (value of y when x = 0) and b is the slope: b = change in y / change in x. The slope measures the rate of change of y per unit change in x and is constant along a straight line.

Understand Variables, Functions, Graphs, the Straight Line and its Slope

Variables and constants. A *variable* can take different values (price, quantity); a *constant* has a fixed value. An *independent variable* is chosen freely; a *dependent variable* depends on it. A *function* states the relationship: y = f(x).

Graphs. Plot x on the horizontal axis and y on the vertical axis. A table of values gives points; join them to draw the graph. In economics the dependent variable is sometimes plotted on the horizontal axis by convention (demand curves show price on the vertical axis although quantity depends on price).

Straight line. The linear function y = a + bx.

*Intercept a:* the value of y when x = 0 (where the line cuts the y-axis).
*Slope b:* the change in y for a one-unit change in x.

Slope. For two points (x1, y1) and (x2, y2): slope = (y2 - y1) / (x2 - x1) = Δy / Δx. A positive slope: the line rises (both variables move together); a negative slope: the line falls; zero slope: a horizontal line (y constant); undefined slope: a vertical line. The slope is the same everywhere on a straight line.

Finding the equation from two points. Find the slope b, then substitute one point into y = a + bx to find a.

Slope as a rate of change. A slope of 4 means y rises by 4 for each unit rise in x. In economics, the slope of the cost line is the marginal cost; of the consumption line, the marginal propensity to consume.

x-intercept. Put y = 0 to find where the line cuts the x-axis.

Non-linear function. The slope changes from point to point (a curve); over an interval, the average rate of change = Δy / Δx.

Variables, Functions, Graphs, the Straight Line and its Slope: key facts

Straight line: y = a + bx | a = intercept (value of y when x = 0), b = slope; Linear function.
Slope: Slope = (y2 - y1) / (x2 - x1) | Change in y over change in x; Same at all points.
Intercepts: y-intercept: put x = 0 | x-intercept: put y = 0; Where it cuts the axes.

How to answer Variables, Functions, Graphs, the Straight Line and its Slope questions

Identify the variables, then find the slope and intercept.

1Write the line as y = a + bx.
2Read a as the y-intercept and b as the slope.
3From two points find b, then a by substitution.
4Use the equation to find y for a given x or vice versa.

Variables, Functions, Graphs, the Straight Line and its Slope in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Slope as a rate

Suppose a taxi charges a fixed fare of Rs 40 plus Rs 15 per kilometre.

Write the fare function and find the fare for 12 km, and say what the slope means.

Try it first, then show the solutionSolution

Fare = 40 + 15x.
For 12 km: 40 + 15 x 12 = Rs 220.
The slope 15 is the additional fare per kilometre (marginal cost).

Answer: Fare = 40 + 15x; Rs 220 for 12 km; slope = Rs 15 per km.

Memory hook: Intercept: starting point. Slope: change per unit.

Tricks to remember Variables, Functions, Graphs, the Straight Line and its Slope

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

Rise over run
Slope = rise (Δy) over run (Δx).
Use it when: Slope.
a is where it starts
a is the value of y when x is zero.
Use it when: Intercept.
Down is negative
A falling line has a negative slope.
Use it when: Sign.

Variables, Functions, Graphs, the Straight Line and its Slope: solved examples

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

Example 1: Slope sign · 1 mark · MCQBoard level

The slope of the line y = 50 - 2x is:
(a) 50 (b) 2 (c) -2 (d) -50

Try it first, then show the solutionSolution

Compare with y = a + bx: b = -2.

Answer: (c) -2

Example 2: Intercept and slope · 3 marks · Short answerBoard level

For the function y = 12 + 3x, find the intercept, the slope, and the values of y at x = 0, 2, 5. (3 marks)

Try it first, then show the solutionSolution

Intercept a = 12; slope b = 3.
At x = 0: 12; at x = 2: 12 + 6 = 18; at x = 5: 12 + 15 = 27.

Answer: Intercept 12; slope 3; y = 12, 18, 27.

Example 3: Equation from two points · 5 marks · Long answerHarder board level

A straight line passes through (2, 10) and (6, 26). Find the slope, the equation, the y value at x = 10 and the x-intercept. (5 marks)

Try it first, then show the solutionSolution

Slope = (26 - 10) / (6 - 2) = 16 / 4 = 4.
Substitute (2, 10): 10 = a + 4 x 2, so a = 2. Equation: y = 2 + 4x.
At x = 10: y = 2 + 40 = 42.
x-intercept: 0 = 2 + 4x, so x = -0.5.

Answer: Slope 4; y = 2 + 4x; y = 42 at x = 10; x-intercept -0.5.

Example 4: Using slope · 5 marks · Long answerHarder board level

The relationship between a firm's cost C and output Q is C = 800 + 25Q. Find the fixed cost, the cost of producing 40 units, the extra cost of 15 more units, and the output when total cost is Rs 2,300. Explain the slope. (5 marks)

Try it first, then show the solutionSolution

Fixed cost (intercept) = Rs 800.
C(40) = 800 + 25 x 40 = Rs 1,800.
Extra cost of 15 units = 25 x 15 = Rs 375.
2,300 = 800 + 25Q, so Q = 1,500 / 25 = 60 units.
The slope 25 is the extra cost of one more unit of output (marginal cost).

Answer: Fixed cost Rs 800; cost of 40 units Rs 1,800; Rs 375 for 15 more; Q = 60 for Rs 2,300; slope = marginal cost Rs 25.

Example 5: Average rate of change on a curve · 4 marks · Short answerHOTS / competency

A non-linear function is y = x squared. Find the average rate of change between x = 2 and x = 3, and between x = 3 and x = 4. Why is a single slope not enough to describe this function? (4 marks)

Try it first, then show the solutionSolution

x = 2, y = 4; x = 3, y = 9; x = 4, y = 16.
Average rate 2 to 3: 5; 3 to 4: 7.
The rate of change keeps rising; a curve has a different slope at every point, unlike a straight line.

Answer: 5 and 7: the slope changes along the curve, so a single slope describes only a straight line.

Where students lose marks in Variables, Functions, Graphs, the Straight Line and its Slope

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

Mistake: Slope as x over y
Inverted ratio.
Wrong working
4/16
Right working
16/4 = 4
Do this: Rise over run.
Usually costs: 2 marks.
Mistake: Subtracting coordinates in different orders
Sign slip.
Wrong working
(26 - 10) / (2 - 6)
Right working
(26 - 10) / (6 - 2)
Do this: Same order for top and bottom.
Usually costs: 2 marks.
Mistake: Reading the slope as the intercept
Mixing a and b.
Wrong working
a = 4
Right working
b = 4, a = 2
Do this: a is the constant term.
Usually costs: 1-2 marks.

Exam tips

Write the general form y = a + bx first.
Use the same order for subtraction in the slope.
Interpret the slope in economic terms.

Practice questions on Variables, Functions, Graphs, the Straight Line and its Slope

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
In the function y = 5 + 2x, the value of y when x = 0 is:
(a) 2 (b) 5 (c) 7 (d) 0
Hint
Intercept.
Answer
(b) 5
Q2 · 3 marks · Short answerBoard level
Find the slope of the line through (1, 7) and (4, 1), and write its equation. (3 marks)
Hint
Negative slope.
Answer
Slope = (1 - 7) / (4 - 1) = -2. From (1, 7): 7 = a - 2, so a = 9. Equation y = 9 - 2x.
Q3 · 5 marks · Long answerHarder board level
The line y = 60 - 4x gives the quantity demanded y at price x. Find the intercepts on both axes, the quantity at price 8, and the price when quantity is 20. Explain the slope. (5 marks)
Hint
x-intercept: y = 0.
Answer
y-intercept 60; x-intercept: 0 = 60 - 4x, x = 15. At price 8: y = 60 - 32 = 28. For y = 20: 20 = 60 - 4x, x = 10. The slope -4: each Rs 1 rise in price lowers quantity by 4 units.
Q4 · 5 marks · Long answerHarder board level
A salesman earns a fixed salary of Rs 12,000 plus Rs 150 per unit sold. A second job pays a fixed Rs 8,000 plus Rs 250 per unit. Write the two earning functions, find the number of units at which earnings are equal, and which job is better for 30 units. (5 marks)
Hint
12,000 + 150x = 8,000 + 250x.
Answer
E1 = 12,000 + 150x; E2 = 8,000 + 250x. Equal when 4,000 = 100x, so x = 40 units (earnings Rs 18,000). For 30 units: E1 = Rs 16,500; E2 = Rs 15,500, so the first job is better below 40 units.
Q5 · 3 marks · Short answerHOTS / competency
A student plots Price on the x-axis and Quantity on the y-axis for the demand function Q = 40 - 2P, and a second student plots Quantity on the x-axis and Price on the y-axis. Explain how the slope and intercept differ in the two graphs. (3 marks)
Hint
Invert the equation: P = 20 - 0.5Q.
Answer
First graph: slope -2 and y-intercept 40. Second graph (P = 20 - 0.5Q): slope -0.5 and y-intercept 20. The line is the same but its slope and intercept depend on which variable is on which axis (the slopes are reciprocals).

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.Straight line equation?Show answer
y = a + bx.
2.Slope formula?Show answer
Δy / Δx.
3.Intercept means?Show answer
Value of y when x is zero.
4.Falling line has?Show answer
Negative slope.
5.Slope of a curve is?Show answer
Different at different points.

Variables, Functions, Graphs, the Straight Line and its Slope: frequently asked questions

Is slope always positive?

No.

What is a constant?

A fixed value.

Does a horizontal line have slope?

Zero.

Why plot a graph?

To see the relationship clearly.