Question 3 – Form Linear Equations and Solve Graphically
📚 What This Question Asks
This question requires us to:
- Read the word problem carefully
- Identify the unknown quantities (variables)
- Form two linear equations based on given conditions
- Solve the equations graphically by plotting both lines
- Find the intersection point (solution)
- Verify the solution algebraically
Part (i) – Mathematics Quiz Problem
Question: 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.
Step 1: Understanding the Problem
Given Information:
- Total students = 10
- Number of girls = Number of boys + 4
To Find:
- Number of boys
- Number of girls
Step 2: Define Variables
Let:
- \( x \) = Number of boys
- \( y \) = Number of girls
Step 3: Form Linear Equations
Equation 1: Total students
Number of boys + Number of girls = 10
\[ x + y = 10 \quad \text{…(1)} \]
Equation 2: Girls are 4 more than boys
Number of girls = Number of boys + 4
\[ y = x + 4 \]
Or rearranging:
\[ x – y = -4 \]Or:
\[ y – x = 4 \quad \text{…(2)} \]Step 4: Find Points for Graphing
For Equation (1): \( x + y = 10 \) or \( y = 10 – x \)
| x (Boys) | 0 | 5 | 10 |
|---|---|---|---|
| y = 10 – x | 10 | 5 | 0 |
| Points (x, y) | (0, 10) | (5, 5) | (10, 0) |
For Equation (2): \( y = x + 4 \)
| x (Boys) | 0 | 3 | 6 |
|---|---|---|---|
| y = x + 4 | 4 | 7 | 10 |
| Points (x, y) | (0, 4) | (3, 7) | (6, 10) |
Step 5: Visual Graph Representation
📊 Graphical Solution
Graph showing intersection at (3, 7) – 3 boys and 7 girls
📊 Graph Analysis:
- Blue Line: \( x + y = 10 \) (Total students = 10)
- Red Line: \( y = x + 4 \) (Girls are 4 more than boys)
- Intersection Point: (3, 7)
- Solution: 3 boys and 7 girls
Step 6: Algebraic Verification
Solving algebraically (Substitution Method):
From equation (2): \( y = x + 4 \)
Substitute in equation (1): \( x + y = 10 \)
\[ x + (x + 4) = 10 \] \[ 2x + 4 = 10 \] \[ 2x = 6 \] \[ x = 3 \]Then: \( y = 3 + 4 = 7 \)
Step 7: Verification
Check in equation (1): \( x + y = 10 \)
LHS = \( 3 + 7 = 10 \) = RHS ✓
Check in equation (2): \( y = x + 4 \)LHS = \( 7 \), RHS = \( 3 + 4 = 7 \) ✓
Logical Check:- Total students = 3 + 7 = 10 ✓
- Girls (7) are 4 more than boys (3) ✓
✅ Answer:
Number of boys = 3
Number of girls = 7
Number of boys = 3
Number of girls = 7
Part (ii) - Pencils and Pens Cost Problem
Question: 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.
Step 1: Understanding the Problem
Given Information:
- 5 pencils + 7 pens = ₹50
- 7 pencils + 5 pens = ₹46
To Find:
- Cost of one pencil
- Cost of one pen
Step 2: Define Variables
Let:
- \( x \) = Cost of one pencil (in ₹)
- \( y \) = Cost of one pen (in ₹)
Step 3: Form Linear Equations
Equation 1: 5 pencils and 7 pens cost ₹50
\[ 5x + 7y = 50 \quad \text{...(1)} \]
Equation 2: 7 pencils and 5 pens cost ₹46
\[ 7x + 5y = 46 \quad \text{...(2)} \]
Step 4: Find Points for Graphing
For Equation (1): \( 5x + 7y = 50 \) or \( y = \frac{50 - 5x}{7} \)
| x (Pencil) | 3 | 10 | -4 |
|---|---|---|---|
| y = (50 - 5x)/7 | 5 | 0 | 10 |
| Points (x, y) | (3, 5) | (10, 0) | (-4, 10) |
For Equation (2): \( 7x + 5y = 46 \) or \( y = \frac{46 - 7x}{5} \)
| x (Pencil) | 3 | 8 | -2 |
|---|---|---|---|
| y = (46 - 7x)/5 | 5 | -2 | 12 |
| Points (x, y) | (3, 5) | (8, -2) | (-2, 12) |
Note: Both lines pass through (3, 5)!
Step 5: Visual Graph Representation
📊 Graphical Solution
Graph showing intersection at (3, 5) - Pencil costs ₹3, Pen costs ₹5
📊 Graph Analysis:
- Blue Line: \( 5x + 7y = 50 \) (5 pencils + 7 pens = ₹50)
- Red Line: \( 7x + 5y = 46 \) (7 pencils + 5 pens = ₹46)
- Intersection Point: (3, 5)
- Solution: Pencil = ₹3, Pen = ₹5
Step 6: Algebraic Verification (Elimination Method)
Solving algebraically:
Equation (1): \( 5x + 7y = 50 \)
Equation (2): \( 7x + 5y = 46 \)
Multiply (1) by 7: \( 35x + 49y = 350 \) ...(1')
Multiply (2) by 5: \( 35x + 25y = 230 \) ...(2')
Subtract (2') from (1'):
\[ 24y = 120 \] \[ y = 5 \]Substitute \( y = 5 \) in equation (1):
\[ 5x + 7(5) = 50 \] \[ 5x + 35 = 50 \] \[ 5x = 15 \] \[ x = 3 \]Step 7: Verification
Check in equation (1): \( 5x + 7y = 50 \)
LHS = \( 5(3) + 7(5) = 15 + 35 = 50 \) = RHS ✓
Check in equation (2): \( 7x + 5y = 46 \)LHS = \( 7(3) + 5(5) = 21 + 25 = 46 \) = RHS ✓
Logical Check:- 5 pencils (₹3 each) + 7 pens (₹5 each) = ₹15 + ₹35 = ₹50 ✓
- 7 pencils (₹3 each) + 5 pens (₹5 each) = ₹21 + ₹25 = ₹46 ✓
✅ Answer:
Cost of one pencil = ₹3
Cost of one pen = ₹5
Cost of one pencil = ₹3
Cost of one pen = ₹5
📊 Summary of Solutions
| Part | Problem | Equations | Solution |
|---|---|---|---|
| (i) | Boys & Girls Quiz | \( x + y = 10 \) \( y = x + 4 \) | Boys = 3 Girls = 7 |
| (ii) | Pencils & Pens | \( 5x + 7y = 50 \) \( 7x + 5y = 46 \) | Pencil = ₹3 Pen = ₹5 |
⚠️ Common Mistakes to Avoid
❌ Mistake 1: Misinterpreting "4 more than" as subtraction instead of addition.
✅ Correct: "Girls are 4 more than boys" means y = x + 4, not y = x - 4.
✅ Correct: "Girls are 4 more than boys" means y = x + 4, not y = x - 4.
❌ Mistake 2: Forgetting to label axes correctly when graphing.
✅ Correct: Always label what each axis represents (boys/girls, pencils/pens, etc.).
✅ Correct: Always label what each axis represents (boys/girls, pencils/pens, etc.).
❌ Mistake 3: Not choosing appropriate scale for the graph.
✅ Correct: Choose a scale that fits all important points on your graph paper.
✅ Correct: Choose a scale that fits all important points on your graph paper.
❌ Mistake 4: Reading intersection point incorrectly from the graph.
✅ Correct: Always verify graphical solution algebraically.
✅ Correct: Always verify graphical solution algebraically.
💡 Key Points to Remember
- Read Carefully: Understand what the problem is asking
- Define Variables: Clearly state what x and y represent
- Form Equations: Translate word statements into mathematical equations
- Choose Scale: Pick appropriate scale for graphing
- Plot Accurately: Use at least 3 points per line for accuracy
- Find Intersection: The solution is where both lines meet
- Verify Algebraically: Always check your graphical solution by solving algebraically
- Check Logically: Does the answer make sense in the context of the problem?
- Units Matter: Include units (₹, students, etc.) in your final answer
📝 Practice Similar Problems
- A father is 3 times as old as his son. After 12 years, he will be twice as old. Find their present ages.
- The sum of two numbers is 25 and their difference is 13. Find the numbers.
- 2 tables and 3 chairs cost ₹700. 3 tables and 2 chairs cost ₹800. Find the cost of each.
- In a class of 40 students, boys are 8 more than girls. Find the number of boys and girls.
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.