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Pair of 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 [[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 fo...

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