Welcome, young mathematicians! In this session, we will dive deep into the solutions for Class 9 Maths Chapter 4, Exercise 4.3 (MP Board / NCERT syllabus). Drawing graphs and interpreting linear equations are fundamental skills not just for your term exams, but also for future competitive examinations. Let's master these concepts step-by-step!
Exercise 4.3 Solutions
(i) x + y = 4
(ii) y = 3x
(i) Equation: x + y = 4
To draw the graph, let's find at least two points (solutions) for this equation:
- If x = 0, then 0 + y = 4 ⇒ y = 4. Point is A(0, 4).
- If y = 0, then x + 0 = 4 ⇒ x = 4. Point is B(4, 0).
| x | 0 | 4 |
|---|---|---|
| y | 4 | 0 |
(ii) Equation: y = 3x
Let's find solutions for this equation:
- If x = 0, then y = 3(0) = 0. Point is O(0, 0) [Origin].
- If x = 1, then y = 3(1) = 3. Point is P(1, 3).
- If x = -1, then y = 3(-1) = -3. Point is Q(-1, -3).
| x | 0 | 1 | -1 |
|---|---|---|---|
| y | 0 | 3 | -3 |
We are given a point (2, 14), which means x = 2 and y = 14.
We can form multiple linear equations that satisfy these values:
- Equation 1: x + y = 16 (since 2 + 14 = 16)
- Equation 2: 7x - y = 0 (since 7(2) - 14 = 14 - 14 = 0)
- Equation 3: y = 7x (since 14 = 7 × 2)
Since the point (3, 4) lies on the graph of the equation 3y = ax + 7, it must satisfy the equation.
Substitute x = 3 and y = 4 in the given equation:
3(4) = a(3) + 7
⇒ 12 = 3a + 7
⇒ 3a = 12 - 7
⇒ 3a = 5
⇒ a = 5/3
Thus, the value of a is 5/3.
Let the total distance covered = x km
Total taxi fare = Rs. y
Fare for the 1st kilometer = Rs. 8
Remaining distance = (x - 1) km
Fare for the remaining distance = Rs. 5 per km
According to the condition:
Total Fare = Fare for 1st km + Fare for remaining distance
y = 8 + 5(x - 1)
y = 8 + 5x - 5
y = 5x + 3
Or, 5x - y + 3 = 0 is the required linear equation.
Table of solutions to draw the graph:
- If x = 0, y = 3. Point is (0, 3).
- If x = 1, y = 8. Point is (1, 8).
- If x = 2, y = 13. Point is (2, 13).
| x (Distance in km) | 0 | 1 | 2 |
|---|---|---|---|
| y (Fare in Rs.) | 3 | 8 | 13 |
ЁЯЪА Aptitude Corner (For NTSE & Olympiads)
Concept: Slope-Intercept Form (y = mx + c)
In competitive exams, linear equations are analyzed using the slope-intercept form. For the equation y = 5x + 3:
- Slope (m) = 5: This tells us the rate of change. For every 1 km increase in distance, the fare increases by Rs. 5.
- Y-intercept (c) = 3: This is the starting value on the y-axis (when x = 0).
Quick Trick: To find the intercept on the x-axis, put y = 0. To find the intercept on the y-axis, put x = 0. This helps in sketching graphs in under 5 seconds!
We hope these step-by-step solutions help clear your doubts about plotting and solving linear equations. Remember, consistent practice is the key to mastering algebra! Keep practicing, and feel free to reach out if you have any questions.
Academic Discussion & Doubts