**Linear Equations in Two Variables**

**Page No: 70**

**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**

**Answer**

Since the equation, y = 3x + 5 is a linear equation in two variables. It will have (iii) infinitely many solutions.

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

- (i) 2x + y = 7
- (ii) πx + y = 9
- (iii) x = 4y

**Answer**

**(i) 2x + y = 7**⇒ y = 7 – 2x

→ Put x = 0,

y = 7 – 2 × 0 ⇒ y = 7

(0, 7) is the solution.

→ Now, put x = 1

y = 7 – 2 × 1 ⇒ y = 5

(1, 5) is the solution.

→ Now, put x = 2

y = 7 – 2 × 2 ⇒ y = 3

(2, 3) is the solution.

→ Now, put x = -1

y = 7 – 2 × -1 ⇒ y = 9

(-1, 9) is the solution.

The four solutions of the equation 2x + y = 7 are (0, 7), (1, 5), (2, 3) and (-1, 9).

**(ii) πx + y = 9**⇒ y = 9 – πx

→ Put x = 0,

y = 9 – π×0 ⇒ y = 9

(0, 9) is the solution.

→ Now, put x = 1

y = 9 – π×1 ⇒ y = 9-π

(1, 9-π) is the solution.

→ Now, put x = 2

y = 9 – π×2 ⇒ y = 9-2π

(2, 9-2π) is the solution.

→ Now, put x = -1

y = 9 – π× -1 ⇒ y = 9+π

(-1, 9+π) is the solution.

The four solutions of the equation πx + y = 9 are (0, 9), (1, 9-π), (2, 9-2π) and (-1, 9+π).

**(iii) x = 4y**→ Put x = 0,

0 = 4y ⇒ y = 0

(0, 0) is the solution.

→ Now, put x = 1

1 = 4y ⇒ y = 1/4

(1, 1/4) is the solution.

→ Now, put x = 4

4 = 4y ⇒ y = 1

(4, 1) is the solution.

→ Now, put x = 8

8 = 4y ⇒ y = 2

(8, 2) is the solution.

The four solutions of the equation πx + y = 9 are (0, 0), (1, 1/4), (4, 1) and (8, 2).

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

- (i) (0, 2)
- (ii) (2, 0)
- (iii) (4, 0)
- (iv) (√2, 4√2)
- (v) (1, 1)

**Answer**

**(i) Put x = 0 and y = 2 in the equation x – 2y = 4.**0 – 2×2 = 4

⇒ -4 ≠ 4

∴ (0, 2) is not a solution of the given equation.

**(ii) Put x = 2 and y = 0 in the equation x – 2y = 4.**2 – 2×0 = 4

⇒ 2 ≠ 4

∴ (2, 0) is not a solution of the given equation.

**(iii) Put x = 4 and y = 0 in the equation x – 2y = 4.**4 – 2×0 = 4

⇒ 4 = 4

∴ (4, 0) is a solution of the given equation.

**(iv) Put x = √2 and y = 4√2 in the equation x – 2y = 4.**√2 – 2×4√2 = 4 ⇒ √2 – 8√2 = 4 ⇒ √2(1 – 8) = 4

⇒ -7√2 ≠ 4

∴ (√2, 4√2) is not a solution of the given equation.

**(v) Put x = 1 and y = 1 in the equation x – 2y = 4.**1 – 2×1 = 4

⇒ -1 ≠ 4

∴ (1, 1) is not a solution of the given equation.

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

**Answer**

Given equation = 2x + 3y = k

x = 2, y = 1 is the solution of the given equation.

A/q,

Putting the value of x and y in the equation, we get

2×2 + 3×1 = k

⇒ k = 4 + 3

⇒ k = 7