Pair of Linear Equations in Two Variables

Question 1

Algebraic Expressions

Answer 1

Algebraic expressions are made of numbers, symbols and the basic arithmetical operations

Question 2

Algebraic Equations and Types

Answer 2

An equation is a sentence in which there is an equality sign between two algebraic expressions.

For example, 2 x + 5 = x + 3, 3 y – 4 = 20 and 5 x + 6 = x + 1 are some examples of equations. Here x and y are unknown quantities and 5, 3, 20, etc., are known quantities.

Question 3

Types of Algebraic Equations

Answer 3

There are 2 types of Algebraic Equations:

     1. Linear Equation

     2. Simple Equation

Question 4

Linear Equation

Answer 4

A linear equation is an equation in which the highest power of the variable is always 1.

Question 5

Simple Equation

Answer 5

A linear equation which has only one unknown is a simple equation. 3 x + 4 = 16 and 2 x – 5 = x + 3 are examples of simple linear equations.

Question 6

Transposition

Answer 6

When a term is moved (transposed) from one side of the equation to the other side, the sign is changed. The positive sign is changed to the negative sign and multiplication is changed to division. Moving a term from one side of the equation to the other side is called transposition. Thus solving a linear equation, in general, comprises two kinds of transposition

Question 7

Simultaneous Linear Equations

Answer 7

Very often we come across equations involving more than one unknown. In such cases we require more than one condition or equation. Generally, when there are two unknowns, we require two equations to solve the problem. When there are three unknowns, we require three equations and so on.

Question 8

Number of Ways a Linear Equation with 2 variable be solved

Answer 8

When two equations, each in two variables, are given, they can be solved in five ways.

1. Elimination by cancellation
2. Elimination by substitution
3. Adding the two equations and subtracting one equation from the other
4. Cross-multiplication method
5. Graphical method

Question 9

Cross-Multiplication Method

Answer 9

x y 1

———- = ———— = ———–

b1 c2 | c1 a2 | a1 b2

-b2 c1 | -c2 a1 | -a2 b1

(where ; a1​x+b1​y=c1​

         a2​x+b2​y=c2​ 

are 2 linear equation)

Question 10

Nature of Solutions

Answer 10

When we try to solve a pair of equations, we could arrive at three possible results. They are:
1. having a unique solution.
2. having an infinite number of solutions.
3. having no solution.

Question 11

A pair of equations having a unique solution:

Answer 11

a₁/a₂ ≠ b₁/b₂

Question 12

A pair of equations having infinite solutions:

Answer 12

a₁/a₂ = b₁/b₂ = c₁/c₂

Question 13

A pair of equations having no solution at all:

Answer 13

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Question 14

Linear Inequality

Answer 14

If a is any real number, then a is either positive or negative or zero. When a is positive, we write a > 0, which is read as ‘ a is greater than zero’. When a is negative, we write a < 0, which is read as ‘ a is less than zero’. If a is zero, we write a = 0 and in this case, a is neither positive nor negative. The two signs > and < are called the ‘signs of inequalities’

Question 15

Inequality Notation:

Answer 15

1. ’ > ‘ denotes ‘greater than’.
2. ’ < ‘ denotes ‘less than’.
3. ’ ≥ ‘ denotes ‘greater than or equal to’.
4. ’ ≤ ‘ denotes ‘less than or equal to’.

Question 16

Inequation

Answer 16

An open sentence which consists of one of the symbols viz., > , < , ≥ , ≤ is called an inequation.

Question 17

Law of Trichotomy

Answer 17

For any two real numbers a and b , either a > b or a < b or a = b

Question 18

Transitive Property

Answer 18

If a > b and b > c, then a > c

Question 19

What does it mean when a is greater than b?

Answer 19

a is greater than b when a - b is positive, i.e., a > b when a - b > 0.

Question 20

What does it mean when a is less than b?

Answer 20

a is less than b when a - b is negative, i.e., a < b when a - b < 0.

Question 21

If a > b, what can we say about b?

Answer 21

If a > b, then b < a.

Question 22

What is the transitive property of inequalities?

Answer 22

If a > b and b > c, then a > c.

Question 23

If a > b, what can we say about a + c and b + c?

Answer 23

If a > b, then a + c > b + c.

Question 24

If a > b and c > 0, what can we say about ac and bc?

Answer 24

If a > b and c > 0, then ac > bc.

Question 25

If a > b and c < 0, what can we say about ac and bc?

Answer 25

If a > b and c < 0, then ac < bc.

Question 26

What can we say about the square of any real number?

Answer 26

The square of any real number is always greater than or equal to 0.

Question 27

If a > 0, what can we say about -a?

Answer 27

-a < 0.

Question 28

If a > b, what can we say about -a and -b?

Answer 28

If a > b, then -a < -b.

Question 29

If a and b are positive numbers and a > b, what can we say about 1/a and 1/b?

Answer 29

If a and b are positive numbers and a > b, then 1/a < 1/b.

Question 30

If a and b are negative numbers and a > b, what can we say about 1/a and 1/b?

Answer 30

If a and b are negative numbers and a > b, then 1/a < 1/b.

Question 31

If a > 0 and b < 0, what can we say about 1/a and 1/b?

Answer 31

If a > 0 and b < 0, then 1/a > 1/b.

Question 32

Properties of Inequities

Answer 32

For any two non-zero real numbers a and b , * a is said to be greater than b when a − b is positive, i.e., a > b when a − b > 0 and * a is said to be less than b when a − b is negative. i.e., a < b when a − b < 0. Listed below are some properties/results which are needed to solve problems on inequalities. The letters a , b , c , d , etc. represent real numbers. * For any two real numbers a and b , either a > b or a < b or a = b . This property is called Law of Trichotomy. * If a > b , then b < a . * If a > b and b > c, then a > c . This property is called transitive property. * If a > b then a + c > b + c . * If a > b and c > 0, then ac > bc . * If a > b and c < 0, then ac < bc . * If a > b and c > d, then a + c > b + d . * The square of any real number is always greater than or equal to 0. * If a > 0, then − a < 0 and if a > b , then − a < − b . * If a and b are positive numbers and a > b , then 1/ a < 1/ b . * If a and b are negative numbers and a > b , then 1/ a < 1/ b . * If a > 0 and b < 0, then 1/ a > 1/ b

Question 33

Continued inequation

Answer 33

Two inequations of the same type (i.e., both consisting of > or ≥ or both consisting of < or ≤ ) can be combined into a continued inequation as explained below.

Question 34

Linear Inequations

Answer 34

An inequation in which the highest degree of the variables present is one is called a linear inequation

Question 35

Absolute Value (Modulus)

Answer 35

If a is any real number, then, (i) | x | ≤ a ⇒ - a ≤ x ≤ a (ii) | x | ≥ a ⇒ x ≥ a or x ≤ - a

Question 36

Properties of Modulus

Answer 36

Properties of modulus: (1) x = 0 ⇔ | x | = 0; (2) For all values of x , | x | ≥ 0 and -| x | ≤ 0 (3) For all values of x , | x + y | ≤ | x | + | y | (4) || x | − | y || ≤ | x − y | (5) − | x | ≤ x ≤ | x | (6) | xy | = | x || y | (7) x y = x y , y ≠ 0 (8) | x^2 | = x^ 2

Question 37

Interval Notation

Answer 37

We have seen above that the solution set or the range of values that satisfy inequalities are not discrete. Instead, we have, a continuous range of values. Such ranges can be represented using the interval notation. The set of all real numbers between a and b (where a < b ) excluding a and b is represented as ( a , b ) read as "the open interval a , b ".[ a , b ] read as "the closed interval a , b " means all real numbers between a and b including a and b ( a < b ). [ a , b ) means all numbers between a and b , with a being included and b excluded ( a < b ).