## CBSE Class 10 Maths Notes Chapter 3 Pair of Linear equations in Two Variables

CBSE Class 10 Maths Notes for Chapter 3 – Pair of Linear Equations in Two Variables:

Introduction:

- A linear equation in two variables can be represented in the form ax + by = c where a, b, and c are constants and x and y are variables.
- A pair of linear equations in two variables can be represented as: a1x + b1y = c1 a2x + b2y = c2
- The solution of a pair of linear equations is a pair of values of x and y which satisfies both the equations simultaneously.

Methods to Solve a Pair of Linear Equations:

- Graphical Method:

- We plot the two lines representing the two equations on the same graph and the solution is the point of intersection of the two lines.

- Algebraic Method: (i) Substitution Method:

- We solve one equation for one variable and substitute the value obtained in the second equation to get the value of the other variable. (ii) Elimination Method:
- We multiply one or both the equations by suitable constants such that the coefficients of one variable are equal in both the equations, and then subtract one equation from the other to get the value of one variable. This value can be substituted in either of the two equations to obtain the value of the other variable. (iii) Cross-Multiplication Method:
- We multiply the first equation by the coefficient of y in the second equation and the second equation by the coefficient of y in the first equation. Then, we subtract the two equations to obtain a quadratic equation in one variable which can be solved to obtain the values of x and y.

Applications of Linear Equations:

- Linear equations are used to solve problems related to cost, profit, revenue, distance, speed, and time.

Important Formulas:

- Slope of a line = (y2 – y1) / (x2 – x1)
- Point-slope form of a line: y – y1 = m(x – x1)
- Slope-intercept form of a line: y = mx + b
- Standard form of a linear equation: ax + by = c

Summary:

- A pair of linear equations in two variables can be solved using graphical or algebraic methods.
- The slope-intercept and standard forms of a linear equation are commonly used to represent linear equations.
- Linear equations are used in a variety of real-life applications such as cost, revenue, and distance.

**CBSE Class 10 Maths Important Questions Chapter 3 Pair of Linear equations in Two Variables**

- Solve the following pair of linear equations by the substitution method: 2x + y = 5 x – y = 1
- Solve the following pair of linear equations by the elimination method: 3x + 2y = 8 2x – 3y = 1
- Find the values of x and y for the following pair of linear equations: 3x + 2y = 12 2x – 3y = 1
- Find the value of k for which the following pair of linear equations has no solution: 2x + 3y = 7 4x + ky = 11
- Find the value of p and q for the following pair of linear equations: px + qy = 4 2px + 2qy = 8
- A man and his son together have 45 marbles. The man has 5 more marbles than his son. Find the number of marbles each of them has.
- The sum of two consecutive odd numbers is 56. Find the two numbers.
- A train travels 360 km at a uniform speed. If the speed had been 5 km/h less, it would have taken 1 hour more for the same journey. Find the speed of the train.
- A sum of money is to be divided among A, B, and C in the ratio 2:3:5. If the total amount is Rs. 3000, find the share of each.
- The cost of 2 kg of sugar and 1 kg of wheat is Rs. 60. The cost of 1 kg of sugar and 2 kg of wheat is Rs. 70. Find the cost of 1 kg of each.

Note: These questions are important for CBSE Class 10 Maths board exam and can also help in preparing for competitive exams.

**CBSE Class 10 Maths Important Questions Answers Chapter 3 Pair of Linear equations in Two Variables**

- Solve the following pair of linear equations by the substitution method: 2x + y = 5 x – y = 1

Solution: From the second equation, we get: x = y + 1

Substituting the value of x in the first equation, we get: 2(y + 1) + y = 5 3y + 2 = 5 3y = 3 y = 1

Substituting the value of y in x = y + 1, we get: x = 2

Therefore, the solution to the given pair of linear equations is (2,1).

- Solve the following pair of linear equations by the elimination method: 3x + 2y = 8 2x – 3y = 1

Solution: Multiplying the first equation by 3 and the second equation by 2, we get: 9x + 6y = 24 4x – 6y = 2

Adding the above equations, we get: 13x = 26 x = 2

Substituting the value of x in the second equation, we get: 2(2) – 3y = 1 4 – 3y = 1 3y = 3 y = 1

Therefore, the solution to the given pair of linear equations is (2,1).

- Find the values of x and y for the following pair of linear equations: 3x + 2y = 12 2x – 3y = 1

Solution: Multiplying the first equation by 3 and the second equation by 2, we get: 9x + 6y = 36 4x – 6y = 2

Adding the above equations, we get: 13x = 38 x = 38/13

Substituting the value of x in the second equation, we get: 2(38/13) – 3y = 1 76/13 – 3y = 1 3y = 76/13 – 1 3y = 63/13 y = 21/13

Therefore, the solution to the given pair of linear equations is (38/13,21/13).

- Find the value of k for which the following pair of linear equations has no solution: 2x + 3y = 7 4x + ky = 11

Solution: For the given pair of linear equations to have no solution, the slopes of the two lines must be equal and their y-intercepts must be different.

The slope of the first line is -2/3, and the slope of the second line is -4/k. Equating these slopes, we get: -2/3 = -4/k k = 6

Substituting k = 6 in the second equation, we get: 4x + 6y = 11

The y-intercepts of the two lines are 7/3 and 11/6, which are different. Therefore, the value of k for which the given pair of linear equations has no solution is 6.

- Find the value of p and q for the following pair of linear equations: px + qy = 4 2px + 2qy = 8

Solution: Dividing the second equation by 2, we get: px + qy = 4

Comparing the above equation with the first equation, we get: p = 1 and q = 4

Therefore, the value of p is 1 and the value of q is 4.

**CBSE Class 10 Maths Important Questions Answers MCQs Chapter 3 Pair of Linear equations in Two Variables**

The pair of linear equations, 2x – 3y = 4 and 3x + 2y = 5, has

a) unique solution

b) infinite solutions

c) no solution

d) None of these

Answer: a) unique solution

The value of k for which the system of linear equations 2x + y = 5 and x + ky = 3 has no solution is

a) 2

b) 3

c) 5

d) 1

Answer: d) 1

The pair of linear equations, x + y = 4 and x – y = 2, has

a) unique solution

b) infinite solutions

c) no solution

d) None of these

Answer: a) unique solution

The pair of linear equations, 3x + 2y = 5 and 6x + 4y = 10, has

a) unique solution

b) infinite solutions

c) no solution

d) None of these

Answer: b) infinite solutions

If a pair of linear equations is consistent, then it has

a) unique solution

b) infinite solutions

c) no solution

d) None of these

Answer: a) unique solution

## CBSE Class 10 Maths Notes Chapter 3 Pair of Linear equations in Two Variables

- For any linear equation, each solution (x, y) corresponds to a point on the line. General form is given by ax + by + c = 0.
- The graph of a linear equation is a straight line.
- Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is: a
_{1}x + b_{1}y + c_{1}= 0; a_{2}x + b_{2}y + c_{2}= 0

where a_{1}, a_{2}, b_{1}, b_{2}, c_{1}and c_{2}are real numbers, such that a_{1}^{2}+ b_{1}^{2}≠ 0, a_{2}^{2}+ b_{2}^{2}≠ 0. - A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.
- A pair of linear equations in two variables can be represented and solved, by

(i) Graphical method

(ii) Algebraic method

**(i) Graphical method.** The graph of a pair of linear equations in two variables is presented by two lines.**(ii) Algebraic methods.** Following are the methods for finding the solutions(s) of a pair of linear equations:

- Substitution method
- Elimination method
- Cross-multiplication method.

- There are several situations which can be mathematically represented by two equations that are not linear to start with. But we allow them so that they are reduced to a pair of linear equations.
**Consistent system.**A system of linear equations is said to be consistent if it has at least one solution.**Inconsistent system.**A system of linear equations is said to be inconsistent if it has no solution.

**CONDITIONS**** FOR CONSISTENCY**

Let the two equations be:

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

Then,

Relationship between coeff. or the pair of equations | Graph | Number of Solutions | Consistency of System |

a1a2≠b1b2 | Intersecting lines | Unique solution | Consistent |

a1a2=b1b2≠c1c2 | Parallel lines | No solution | Inconsistent |

a1a2=b1b2=c1c2 | Co-incident lines | Infinite solutions | Consistent |

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