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Linear Equations in Two Variables

Introduction to Linear Equations In Two Variables

Recalling Linear Equations in One Variable

We know that algebraic expressions are those that have a few numbers, letters and operators. For example, 2x, 3y + 4 and are all algebraic expressions and the letters x, y and z are the variables in the expressions.

If an algebraic expression is used for equating two different values or expressions, then it becomes an equation. For example, 2x = 4, 3y + 4 = 2y and are all equations.

Now, consider the equation 2x = 4. It has only one variable term, i.e., 2x. The exponent of variable x is 1 and this is the highest exponent in the equation. We know that an equation having the highest exponent as 1 is known as a linear equation; so, 2x = 4 is a linear equation. Also, since the equation has only one variable x, it is a linear equation in one variable. Similarly, 3y + 4 = 2y and are also linear equations in one variable.

There are also equations having more than one variable. In this lesson, we will learn about linear equations in two variables.

Introduction to Linear Equations in Two Variables

A linear equation comprising two different variables is called a linear equation in two variables. Let us consider the equation. This equation is used to compare the temperatures on the Celsius (C) and Fahrenheit (F) scales.

In the equation, C and F are both variables; thus, it is an equation in two variables. Also, the degree of the equation is 1, so it is a linear equation in two variables.

Other examples of linear equations in two variables: 3x − 4y = 4, and

The general form of a linear equation in two variables is ax + by + c = 0. Here, x and y are variables while a, b and c are constants.

Concept Builder

The highest exponent of a variable involved in an equation is the degree of that equation.

For example, in the equation 3y + 4 = 2y, the highest exponent of variable y is 1; so, the degree of the equation is 1, or we can say that it is a first-degree equation.

Did You Know?

−40° is the only point at which the Celsius and Fahrenheit scales coincide. So, −40°C = −40°F

Solved Examples

Easy

Example:

Identify the linear equations in two variables among the following equations.

i)

ii)

iii)

iv)

v)

Solution:

i) Since the equation consists of only one variable x, it is not a linear equation in two variables.

ii) The equation can be reduced to the general form of a linear equation in two variables, i.e., ax + by + c = 0.

The equation is a first-degree equation and consists of two variables t and D. Thus, it is a linear equation in two variables.

iii) The equation consists of two variables x and y, but its degree is 2. Hence, it is not a linear equation in two variables.

iv) The equation can be reduced to the general form of a linear equation in two variables, i.e., ax + by + c = 0.

⇒ 34x = 6y

⇒ 34x − 6y = 0

The equation is a first-degree equation and consists of two variables x and y. Thus, it is a linear equation in two variables.

v) The equation consists of three variables x, t and z; so, it is not a linear equation in two variables.

Linear Equations in Real Life

Let us try to find Madhuri’s age from what she has just told Arya.

We know that Madhuri’s age is 12 less than twice Arya’s age. So, we have two values to equate: ‘Madhuri’s age’ and ‘12 less than twice Arya’s age’.

Madhuri’s age = 12 less than twice Arya’s age

⇒ Madhuri’s age = Twice Arya’s age − 12

Here, Madhuri’s age and Arya’s age are two quantities that are related to each other and can change in value; so, we can represent them by using variables.

Thus, you can see how we need to form equations to solve such problems from our daily life. In this lesson, we will learn to represent real-life situations as linear equations.

Expressing Situations as Linear Equations in Two Variables

Sunita and Priyanka go shopping. On their way, Priyanka tells Sunita, ‘I have Rs 150 more than two-fifth the amount you have.’ Let us represent this situation mathematically.

As per what Priyanka says, if Sunita has Rs 200, then Priyanka has Rs 150 more than two-fifth of Rs 200, i.e., Rs= Rs 230. Similarly, if Sunita has Rs 500 then Priyanka has Rs= Rs 350.

So, the amount that Priyanka has depends upon the amount that Sunita has. Here, we have two unknown values, i.e., ‘the amount with Priyanka’ and ‘the amount with Sunita’. Let us suppose that Priyanka has Rs x and Sunita has Rs y.

Now, two-fifth the amount with Sunita = Rs

Rs 150 more than the two-fifth the amount with Sunita =

According to the given condition, we have:

The amount with Priyanka = Rs 150 more than two-fifth the amount with Sunita

This linear equation in the two variables x and y is the mathematical representation of the given situation. Similarly, other real-life situations can be expressed as linear equations.

Expressing Situations as Linear Equations in Two Variables

Solved Examples

Easy

Example 1:

Express the following situations as linear equations in two variables.

i) The cost of three DVDs and five CDs is Rs 400.

ii) The perimeter of a rectangula…

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