Graphical Method of Linear Equations Explained

 

❓ Question

From the pair of linear equations in the following problems, form equations and find their solution graphically.

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(i) Mathematics Quiz Problem

10 students of Class X took part in a Mathematics quiz.
If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

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✍️ Short Concept

To solve word problems using pair of linear equations:

✅ Define variables
✅ Form equations from conditions
✅ Solve equations graphically or algebraically

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🔷 Step 1 — Define Variables

Let:

$$x = \text{number of girls}$$
$$y = \text{number of boys}$$

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🔷 Step 2 — Form Equations

Total students = 10

$$x+y=10$$

Girls are 4 more than boys

$$x-y=4$$

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🔷 Step 3 — Solve Equations

Add both equations:

$$x+y+x-y=10+4$$
$$2x=14$$
$$x=7$$

Substitute in:

$$x+y=10$$
$$7+y=10$$
$$y=3$$

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🔷 Step 4 — Final Answer

$$\boxed{7\ \text{girls}}$$
$$\boxed{3\ \text{boys}}$$

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❓ Question (ii)

5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46.

Find the cost of one pencil and one pen.

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🖼️ Solution Image

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✍️ Short Concept

For cost-based word problems:

✅ Take unknown costs as variables
✅ Form linear equations
✅ Solve simultaneously

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🔷 Step 1 — Define Variables

Let:

$$x = \text{cost of one pencil}$$
$$y = \text{cost of one pen}$$

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🔷 Step 2 — Form Equations

From first condition:

$$5x+7y=50$$

From second condition:

$$7x+5y=46$$

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🔷 Step 3 — Solve Equations

Multiply first equation by 7:

$$35x+49y=350$$

Multiply second equation by 5:

$$35x+25y=230$$

Subtract:

$$24y=120$$
$$y=5$$

Substitute in:

$$5x+7(5)=50$$
$$5x+35=50$$
$$5x=15$$
$$x=3$$

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🔷 Step 4 — Final Answer

$$\boxed{\text{Cost of one pencil} = ₹3}$$
$$\boxed{\text{Cost of one pen} = ₹5}$$

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⭐ Important Concept

For pair of linear equations:

• Intersecting lines → One unique solution
• Parallel lines → No solution
• Coincident lines → Infinite solutions

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📚 Related Topics

• Find Zeroes and Verify Relationship Between Coefficients
• Find Quadratic Polynomial from Sum and Product of Zeroes
• Quadratic Polynomial with Irrational Zeroes Explained