Linear Equations in Two Variables: 9 Math

NCERT Exercise 4.2: 9th math

In this NCERT exercise 4.2 class ninth math, we will learn to find the solution of linear equations in two variables. A linear equation in two variables has infinitely many solutions. Question (1) of this exercise, is to find whether the above statement about the solutions of a linear equation in two variables is true or not. And in the three questions of question (2) of this exercise four solutions of each of the linear equations given are to find.

Solution of NCERT Exercise 4.2 from question (1) and (2)

Question (1) Which 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 (iii) infinitely many solutions

Because "A linear equation in two variables has infinitely many solutions"

Therefore, the correct answer is the option "(iii) infinitely many solutions"

Explanation

For the given linear equation in two variable, y = 3x + 5

(a)If x = 0

∴ y = 3 × 0 + 5

⇒ y = 5

Thus, solution is x = 0 and y = 5, i.e. (0, 5)

And, (b) if x = 1

∴ y = 3 × 1 + 5

⇒ y = 3 + 5

⇒ y = 8

Thus, solution for the given equation is (1, 5)

Similarly, (c) if y = 0

∴ 0 = 3x + 5

⇒ 3x + 5 = 0

⇒ 3x = – 5

⇒ x = ^{– 5}/₃

∴ The solution is (^{– 5}/₃, 0)

In a similar way, by taking different values for x and y, we can get infinitely many solutions for the given linear equation in two variables.

Therefore, option (iii) infinitely many solutions is the correct answer.

Question (2) Write four solutions for each of the following equations
(i) 2x + y = 7
(ii) π x + y = 9
(iii) x = 4 y

Solution of (i) 2x + y = 7

(1) If x = 0

After substituting the value of x = 0 in the given linear equation in two variable, we get

2 × 0 + y = 7

⇒ 0 + y = 7

⇒ y = 7

Thus, solution is (0, 7)

(2) If x = 1

After substituting the value of x = 1 in the given linear equation in two variable, we get

2 × 1 + y = 7

⇒ 2 + y = 7

⇒ y = 7 – 2

⇒ y = 5

Thus, solution is (1, 5)

(3) If x = 2

After substituting the value of x = 2 in the given linear equation in two variable, we get

2 × 2 + y = 7

⇒ 4 + y = 7

⇒ y = 7 – 4

⇒ y = 3

Thus, solution is (2, 3)

Similarly, by taking different values of y, we can get many solutions

(4) If y = 0

After substituting the value of y = 0 in the given linear equation in two variable, 2x + y = 7 we get

2x + 0 = 7

⇒ 2x = 7

⇒ x = ⁷/₂

⇒ x = 3 ¹/₂

Thus, solution is (3 ¹/₂, 0)

Thus, four solutions for the given linear equation in two variables 2x + y = 7, are

(0, 7), (1, 5), (2, 3), and (3 ¹/₂, 0) Answer

Solution of (ii) π x + y = 9

(1) If x = 0

After substituting the value of x = 0 in the given linear equation in two variable, we get

π × 0 + y = 9

⇒ 0 + y = 9

⇒ y = 9

Thus, solution is (0, 9)

(2) If x = 1

After substituting the value of x = 1 in the given linear equation in two variable, we get

π × 1 + y = 9

⇒ π + y = 9

⇒ y = 9 – π

Thus, solution is (1, 9 – π)

(3) If x = 2

After substituting the value of x = 2 in the given linear equation in two variable, we get

π × 2 + y = 9

⇒ 2 π + y = 9

⇒ y = 9 – 2 π

Thus, solution is (2, 9 – 2π)

(4) If x = 3

After substituting the value of x = 3 in the given linear equation in two variable, we get

π × 3 + y = 9

⇒ 3 π + y = 9

⇒ y = 9 – 3 π

Thus, solution is (3, 9 – 3π)

Similarly, by taking different values of y, we can get many solutions, because we know that a linear equation in two variables has infinitely many solutions.

Thus, four solutions for the given linear equation in two variables πx + y = 9 are

(0, 9), (1, 9 – π), (2, 9 – 2π), and (3, 9 – 3π) Answer

Solution of (iii) x = 4y

(1) If x = 0

After substituting the value of x = 0 in the given linear equation in two variable, x = 4y we get

0 = 4y

⇒ 4y = 0

⇒ y = ⁰/₄

⇒ y = 0

Thus, solution is (0, 0)

(2) If x = 1

After substituting the value of x = 1 in the given linear equation in two variable, x = 4y we get

1 = 4y

⇒ 4y = 1

⇒ y = ¹/₄

Thus, solution is (1, ¹/₄)

(3) If x = 2

After substituting the value of x = 2 in the given linear equation in two variable, x = 4y we get

2 = 4y

⇒ 4y = 2

⇒ y = ²/₄

⇒ y = ¹/₂

Thus, solution is (2, ¹/₂)

(4) If x = 3

After substituting the value of x = 3 in the given linear equation in two variable, x = 4y we get

3 = 4y

⇒ 4y = 3

⇒ y = ³/₄

Thus, solution is (3, ³/₄)

Similarly, by taking different values of y, we can get many solutions, because we know that a linear equation in two variables has infinitely many solutions.

Thus, four solutions for the given linear equation in two variables x = 4y are

(0, 0), (1, ¹/₄), (2, ¹/₂), and (3, ³/₄) Answer

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