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Linear Equations in Two Variables

Introduction to Linear Equations in Two Variable

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 [[mn:glossary]]exponent[[/mn:glossary]] 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.

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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. 

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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.

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

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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.

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An Overview

Linear Equations in Real Life

Let us see what Arya and Madhuri are talking about.

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 ...

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