Class 9 Exam  >  Class 9 Notes  >  Extra Documents & Tests for Class 9  >  NCERT Solutions Chapter 4 - Linear Equation In Two Variables (I), Class 9, Maths

NCERT Solutions for Class 9 Maths Chapter 4 - Chapter 4 - Linear Equation In Two Variables (I),

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).

Answer

Let the cost of pen be y and the cost of notebook be x.
A/q,
Cost  of a notebook = twice the pen = 2y.
∴2y = x
⇒ x - 2y = 0
This is a linear equation in two variables to represent this statement.
 

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               

(ii) x - y/5 - 10 = 0                  

(iii) -2x + 3y = 6                

(iv) x = 3y

(v) 2x = -5y                        

(vi) 3x + 2 = 0                        

(vii) y - 2 = 0                     

(viii) 5 = 2x

Answer

(i) 2x + 3y = 9.35
⇒ 2x + 3y - 9.35 = 0
On comparing this equation with ax + by + c = 0, we get
a = 2x, b = 3 and c = -9.35

(ii) x - y/5 - 10 = 0
On comparing this equation with ax + by + c = 0, we get
a = 1, b = -1/5 and c = -10

(iii) -2x + 3y = 6
⇒ -2x + 3y - 6 = 0
On comparing this equation with ax + by + c = 0, we get
a = -2, b = 3 and c = -6

(iv) x = 3y
⇒ x - 3y = 0
On comparing this equation with ax + by + c = 0, we get
a = 1, b = -3 and c = 0

(v) 2x = -5y
⇒ 2x + 5y = 0
On comparing this equation with ax + by + c = 0, we get
a = 2, b = 5 and c = 0

(vi) 3x + 2 = 0
⇒ 3x + 0y + 2 = 0
On comparing this equation with ax + by + c = 0, we get
a = 3, b = 0 and c = 2

(vii) y - 2 = 0
⇒ 0x + y - 2 = 0
On comparing this equation with ax + by + c = 0, we get
a = 0, b = 1 and c = -2

(viii) 5 = 2x
⇒ -2x + 0y + 5 = 0
On comparing this equation with ax + by + c = 0, we get
a = -2, b = 0 and c = 5

Exercise 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

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
⇒ 2 = 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

The document NCERT Solutions for Class 9 Maths Chapter 4 - Chapter 4 - Linear Equation In Two Variables (I), is a part of the Class 9 Course Extra Documents & Tests for Class 9.
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FAQs on NCERT Solutions for Class 9 Maths Chapter 4 - Chapter 4 - Linear Equation In Two Variables (I),

1. What are linear equations in two variables?
Ans. Linear equations in two variables are algebraic equations that involve two variables, typically represented as x and y, and have a degree of 1. These equations can be written in the form ax + by = c, where a, b, and c are constants. The solutions to these equations are ordered pairs (x, y) that satisfy the equation when substituted into it.
2. How do you solve a linear equation in two variables graphically?
Ans. To solve a linear equation in two variables graphically, we first plot the given equation on a coordinate plane. The graph of the equation is a straight line. Then, we find the point of intersection between this line and the x-axis or y-axis. The coordinates of this point of intersection give the solution to the equation. If the line is parallel to the x-axis or y-axis, the equation has no solution.
3. Can a linear equation in two variables have infinitely many solutions?
Ans. Yes, a linear equation in two variables can have infinitely many solutions. This occurs when the equation represents a line that overlaps with the entire coordinate plane. In such cases, any point on the line can be considered a solution to the equation. These equations are called consistent and dependent systems.
4. How many methods are there to solve linear equations in two variables?
Ans. There are three main methods to solve linear equations in two variables: 1. Graphical method: This method involves plotting the equation on a coordinate plane and finding the point of intersection. 2. Substitution method: In this method, one variable is expressed in terms of the other variable in one equation and then substituted into the other equation to find the value of the remaining variable. 3. Elimination method: In this method, the coefficients of one variable in both equations are manipulated to make them equal or additive inverses. Then, the equations are added or subtracted to eliminate one variable and solve for the other.
5. Can a linear equation in two variables have no solution?
Ans. Yes, a linear equation in two variables can have no solution. This occurs when the equation represents parallel lines on the coordinate plane. Since parallel lines never intersect, there is no point of intersection and hence no solution exists. These equations are called inconsistent and independent systems.
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