Class 10 Maths · Chapter 3 · Topic 1 of 4

Two clues become two equations

Translate two clues using the same two unknowns.

4 connected ideas1 question to try

Ideas you’ll use

Pair of linear equations: Two equations of the form ax + by + c = 0, with a and b not both zero. A common solution must satisfy both equations.

Understand it, one idea at a time

Choose an idea. Follow its reason, calculation and clue in the picture.

Every point fits one bill

Pen price (₹)

Notebook price (₹)

2x + y = 46 · Original teaching example; exact values are available below.
x + 2y = 32 · Original teaching example; exact values are available below.

1. Both bills fit · (20, 6)

Assumptions: Lines join calculated sample points. Curves are visual approximations; use the formula and marked exact values for proof.

Read the plotted values

2x + y = 46

Notebook price (₹)	Pen price (₹)
0	46
0.6	44.8
1.2	43.6
1.8	42.4
2.4	41.2
3	40
3.6	38.8
4.2	37.6
4.8	36.4
5.4	35.2
6	34
6.6	32.8
7.2	31.6
7.8	30.4
8.4	29.2
9	28
9.6	26.8
10.2	25.6
10.8	24.4
11.4	23.2
12	22
12.6	20.8
13.2	19.6
13.8	18.4
14.4	17.2
15	16
15.6	14.8
16.2	13.6
16.8	12.4
17.4	11.2
18	10
18.6	8.8
19.2	7.6
19.8	6.4
20.4	5.2
21	4
21.6	2.8
22.2	1.6
22.8	0.4
23.4	-0.8
24	-2

x + 2y = 32

Notebook price (₹)	Pen price (₹)
0	16
0.6	15.7
1.2	15.4
1.8	15.1
2.4	14.8
3	14.5
3.6	14.2
4.2	13.9
4.8	13.6
5.4	13.3
6	13
6.6	12.7
7.2	12.4
7.8	12.1
8.4	11.8
9	11.5
9.6	11.2
10.2	10.9
10.8	10.6
11.4	10.3
12	10
12.6	9.7
13.2	9.4
13.8	9.1
14.4	8.8
15	8.5
15.6	8.2
16.2	7.9
16.8	7.6
17.4	7.3
18	7
18.6	6.7
19.2	6.4
19.8	6.1
20.4	5.8
21	5.5
21.6	5.2
22.2	4.9
22.8	4.6
23.4	4.3
24	4

The marked intersection gives notebook price ₹20 and pen price ₹6. The rest of each line fits only that bill.

IDEA 1 OF 4

Name the unknowns

Let x be a notebook price and y be a pen price, both in rupees.

x: notebook; y: pen

In the picture: Read the axes as the same prices in both bills.

Read all the reasoning together

Name the unknowns
Let x be a notebook price and y be a pen price, both in rupees.
x: notebook; y: pen
Write each bill
Count the items and multiply by their unknown prices.
2x+y=46; x+2y=32
Require both clues
The meeting pair fits both bills, not just one.
x=20, y=6
Check with units
Put the pair into both originals and state the prices.
2(20)+6=46; 20+2(6)=32

The idea to keep

Translate two clues using the same two unknowns.

Worked example: Check both bills

x = 20, y = 6: 2(20) + 6 = 46; 20 + 2(6) = 32

A pair that fits just one bill is not enough. Include units when stating the answer.

Read the supporting explanation

Let x and y be the prices of a notebook and a pen in rupees. If two notebooks and one pen cost ₹46, then 2x + y = 46. If one notebook and two pens cost ₹32, then x + 2y = 32. The variables mean the same thing in both equations.

Go deeper: Set sensible limits

A price cannot be negative; a count must be a whole number. These conditions belong to the story, even when the equations allow other values.

Your turn

Try the idea yourself. Show your reasoning, then compare.

Question

Two numbers total 17 and the first is 5 more than the second. Form the equations.

Write your own answer and compare it with ours. It’s free.

Pair of Linear Equations in Two Variables Class 10 Notes

Chapter notes

Two clues become two equations

Ideas you’ll use

Understand it, one idea at a time

Every point fits one bill

Name the unknowns

Name the unknowns

Write each bill

Require both clues

Check with units

Your turn

Keep reading Pair of Linear Equations in Two Variables, free