Welcome, students! Today we are diving into Chapter 4: Linear Equations in Two Variables (Exercise 4.1) of the Class 9 Mathematics syllabus, tailored for MP Board and NCERT guidelines. In this session, we will learn how to formulate linear equations from real-life scenarios and represent them in the standard general form: ax + by + c = 0. Let's make mathematics simple and enjoyable!
Exercise 4.1 Solutions
Q1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
(Take the cost of a notebook to be ₹ x and that of a pen to be ₹ y)
Solution:
Let the cost of a notebook be ₹ x.
Let the cost of a pen be ₹ y.
According to the given condition:
Cost of notebook = 2 × Cost of pen
Therefore, we can write the equation as:
x = 2y
Expressing this in the standard form (ax + by + c = 0):
x - 2y = 0
Q2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:
(i) 2x + 3y = 9.35̅
Solution:
Rearranging the equation to the form ax + by + c = 0:
2x + 3y - 9.35̅ = 0
Comparing this with the general form, we get:
a = 2, b = 3, c = -9.35̅
(ii) x - y/5 - 10 = 0
Solution:
The equation is already in the form ax + by + c = 0. It can be written as:
1x + (-1/5)y + (-10) = 0
Comparing with the general form, we get:
a = 1, b = -1/5, c = -10
(iii) -2x + 3y = 6
Solution:
Rearranging the terms:
-2x + 3y - 6 = 0
Comparing with the general form, we get:
a = -2, b = 3, c = -6
(iv) x = 3y
Solution:
Rearranging the terms:
1x - 3y + 0 = 0
Comparing with the general form, we get:
a = 1, b = -3, c = 0
(v) 2x = -5y
Solution:
Rearranging the terms:
2x + 5y + 0 = 0
Comparing with the general form, we get:
a = 2, b = 5, c = 0
(vi) 3x + 2 = 0
Solution:
Since there is no y term, we can write it as 0y:
3x + 0y + 2 = 0
Comparing with the general form, we get:
a = 3, b = 0, c = 2
(vii) y - 2 = 0
Solution:
Since there is no x term, we can write it as 0x:
0x + 1y - 2 = 0
Comparing with the general form, we get:
a = 0, b = 1, c = -2
(viii) 5 = 2x
Solution:
Rearranging the equation:
-2x + 0y + 5 = 0 (or 2x + 0y - 5 = 0)
Comparing with the general form, we get:
a = -2, b = 0, c = 5 (or a = 2, b = 0, c = -5)
💡 Aptitude Corner (NTSE / Olympiad Special)
Key Concept: A linear equation in two variables ax + by + c = 0 represents a straight line. For this equation to be valid as a 'two-variable' equation, both a and b cannot be zero at the same time. Mathematically, this is written as: a2 + b2 ≠ 0.
Quick Trick: When converting standard single variable equations like x = k into two variables, always assign a coefficient of 0 to the missing variable (i.e., x + 0y - k = 0). This helps in quickly finding coordinates during graphical plotting!
We hope these solutions assist you in laying a solid foundation for Linear Equations. Consistent practice is the key to mastering algebra. Keep experimenting with coefficients and plotting them for better clarity. Happy learning!
Academic Discussion & Doubts