Chapter 4 : Linear Equations in Two Variables CBSE notes

Linear Equations in Two Variables

Notes

Linear Equations in Two Variables

➥ A linear equation in two variables is an equation of the form ax + by + c = 0.

where x and y are variables, and a, b, and c are constants.

➥ The graph of such an equation is a straight line.

General Form

• ➥ The general form of a linear equation in two variables is ax + by + c = 0.
• Here, x and y are variables, and a, b, and c are constants.

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Exercise and Solutions

EXERCISE 4.1

1. 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:
x = 2y

2. 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 = 9.35
Solution:
2x + 3y - 9.35 = 0
Here, a = 2, b = 3, c = -9.35.

(ii) x - 5y - 10 = 0
Solution:
x - 5y - 10 = 0
Here, a = 1, b = -5, c = -10.

(iii) -2x + 3y = 6
Solution:
-2x + 3y - 6 = 0
Here, a = -2, b = 3, c = -6.

(iv) x = 3y
Solution:
x - 3y = 0
Here, a = 1, b = -3, c = 0.

(v) 2x = -5y
Solution:
2x + 5y = 0
Here, a = 2, b = 5, c = 0.

(vi) 3x + 2 = 0
Solution:
3x + 0*y + 2 = 0
Here, a = 3, b = 0, c = 2.

(vii) y - 2 = 0
Solution:
0*x + y - 2 = 0
Here, a = 0, b = 1, c = -2.

(viii) 5 = 2x
Solution:
2x - 5 = 0
Here, a = 2, b = 0, c = -5.

Exercises 4.2

1. Which one of the following options is true, and why?

y = 3x + 5 has:

(i) a unique solution,

(ii) only two solutions,

(iii) infinitely many solutions.

Solution:
y = 3x + 5
This is a linear equation in two variables, and the graph of this equation is a straight line. A straight line in a plane has infinitely many points, so it has infinitely many solutions.
Hence, the correct option is (iii) infinitely many solutions.

2. Write four solutions for each of the following equations:

(i) 2x + y = 7
Solution:

                  x = 0, y = 7
                  x = 1, y = 5
                  x = 2, y = 3
                  x = 3, y = 1
               

(ii) πx + y = 9
Solution:

                  x = 0, y = 9
                  x = 1, y = 9 - π
                  x = 2, y = 9 - 2π
                  x = 3, y = 9 - 3π
               

(iii) x = 4y
Solution:

                  y = 0, x = 0
                  y = 1, x = 4
                  y = 2, x = 8
                  y = 3, x = 12
               

3. Check which of the following are solutions of the equation x - 2y = 4 and which are not:

(i) (0, 2)
0 - 2(2) = 0 - 4 ≠ 4 (Not a solution)

(ii) (2, 0)
2 - 2(0) = 2 ≠ 4 (Not a solution)

(iii) (4, 0)
4 - 2(0) = 4 (Solution)

(iv) (2, 2)
2 - 2(2) = 2 - 4 ≠ 4 (Not a solution)

(v) (1, 1)
1 - 2(1) = 1 - 2 ≠ 4 (Not a solution)

4. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.

Solution:
2(2) + 3(1) = k
4 + 3 = k
k = 7