Linear Equation in Two Variables: Graphical Method, Videos and Examples

As the name suggests, linear equations in two variables are the equations in which two variables are present. These equations are used to denote many real-life scenarios involving two unknown quantities. Let us study them in detail.

Line Equation in Two Variables

An algebraic equation is a statement of equality of algebraic expressions involving one or more variables. For example, $$2x-3y+4z=7$$

Linear equation in one variable: An equation with linear expressions in one variable only is known as a linear equation in one variable. For example, $$5x+8 =9–x $$
Linear equation in two variables: An equation which can be put in the form ax +by +c =0, where a, b and c are real numbers and x,y are variables, is called a linear equation in two variable. For example, $$3s-4t=80$$

How to write a linear equation in one variable into a linear equation in two variable? We can always represent a linear equation in one variable as a linear equation in two variables, by taking the coefficient of the second variable as zero and thus writing the given equation in the form ax +by +c =0

Example 1: Write each of the following equation as an equation in two variables $$2y =3$$

Solution: The above equation can be written as $$0\times x +2y-3=0$$

Example 2: The cost of a ball pen is Rs. 5 less than half of the cost of a fountain pen. Write this statement as a linear equation in two variables.

Solution: here, variable quantities are the cost of ball pen and cost of a fountain pen. So, let the cost of a ball pen =Rs x and the cost of a fountain pen =Rs y. According to the question,

$$Cost \; of \; ball \; pen \; = \; Half \; of \; the \; cost \; of \; a \; fountainpen \; -5 \\ x=\frac { y }{ 2 } -5\\ x=\frac { y-10 }{ 2 } \\ 2x=y-10\\ 2x-y+10=0$$

This is the required linear equation in two variables.

Watch Solved Problems on Linear Equations

.embed-container { position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; max-width: 100%; } .embed-container iframe, .embed-container object, .embed-container embed { position: absolute; top: 0; left: 0; width: 100%; height: 100%; }

Method to Solve Linear Equation in Two Variables

Let the given linear equation in two variables be ax+ by +c, where (a) is not equal to zero and (b) is not equal to zero. Then, for finding its solution, we use the following steps

Put an arbitrary value of x (or y) in the given equation and find the corresponding value of the second variable. Thus we get one pair of the solution of the given equation.
Repeat step 1 for another arbitrary value of x (or y) and get another pair of a solution of given equation.

Let’s understand the above method with the help of some examples.

Let us learn to solve linear equation in one variable here.

Solved Example for You

Example 3: Find any four solutions of the equation- $$4x +3y =12$$

Solution: Given equation is $$4x+3y=12\quad …….\left( 1 \right) $$
On putting x =0 in Equation (1), we get
$$4\times 0+3y=12\\ 3y=12\\ y=4$$
So,(0,4) is a solution of given equation. On putting y=0 in Equation (1), we get
$$4x+3\times 0=12\\ 4x=12\\ x=3$$
So, (3,0) is another solution of given equation. On putting x=1 in Equation (1), we get
$$4\times 1+3y=12\\ 3y=12-4\\ 3y=8\\ y=\frac { 8 }{ 3 } $$
So (1,8/3) is another solution of the given equation. Further, putting x=2 in Equation (1), we get
$$4\times 2+3y=12\\ 3y=12-8\\ 3y=4\\ y=\frac { 4 }{ 3 } $$
So, (2,4/3) is also a solution of the given equation. Hence ,the four solution of given equation are (0,4),(3,0),(1,8/3)and (2,4/3).

Graphical Method to solve a Linear Equation in Two Variables

Example 4: Draw the graph of 2x +y =6 and find the point where the graph intersects Y –axis.

Solution: We have the equation – $$2x +y =6$$

Or, $ \; y=6-2x$

When x=0, then $$y=6-2\times 0\\ y=6$$
And when x =1, then $$y=6-2\times 1\\ y=4$$
When x=2, then $$y=6-2\times 2\\ y=2$$

So we have the following table to draw the graph.

X	0	1	2
Y	6	4	2

Here, we have 3 points A (0, 6), B (1, 4), C (2, 2). Now plot these points on the graph and join them with a straight line. variables

Thus, we get the straight line AC, which represents the required graph of the given linear equation. Also, from the graph, it is clear that the graph intersects the Y-axis at the point A (0, 6).

Solved Questions for You

Question 1: What is the linear equation in two variables?

Answer: Linear equations in two variables refer to the equations that have two variables present in it. We use these equations to indicate many real-life situations relating to two unknown quantities.

Question 2: What are the 3 types of variables?

Answer: As you know that variables are things which keep changing in an experiment. Thus, a variable refers to any factor, trait or condition which may exist in conflicting amounts or kinds. Thus, an experiment consists of generally three types of variables which are independent, dependent and controlled.

Question 3: What are the variables in math?

Answer: A variable refers to a quantity that can change within the context of a mathematical problem or experiment. Generally, we make use of a single letter for representing a variable. The letters x, y, and z are common general symbols that we use for variables

Question 4: What are the dependent variables in research?

Answer: The dependent variable refers to the variable that we measure or test in an experiment. For instance, in a study looking at how coaching affects test scores, the dependent variable is going to be the students’ test scores, since that is what we are measuring.