# Math's: Linear Equations: Formation and Solution, Graphical Method

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## Formation and Solution of Linear Equation in Two Variables

Example- Jagriti and Pragati went to market to purchase pencils and pens. Jagriti purchased 4 pencils and 3 pens and spent ₹50 in total while Pragati spent ₹45 in total and purchased 3 pencils and 3 pens. What is the cost of pencil and pen per unit each?

## Formation of Linear Equations in Two Variables

Linear Equations in two variables are solved by forming two equations at a time

Since, we want to find the cost per unit of pencil and pen, let us assume that the cost of 1 pencil is ₹ and the cost of 1 pen

First Equation for Jagriti’s Purchase-

Cost of pencils

Cost of pens

Since, total cost is we have

First Equation for Pragati’s Purchase-

Cost of pencils

Cost of pens

Since, total cost is we have

The above are a linear equation in two variables and as it is of the form

## Solution of Linear Equations in Two Variables

There are several methods of solving system of two linear equations in two variables. We shall now discuss here two algebraic methods which are-

Substitution Method

Elimination method

### Substitution Method

In this method, we find the value of one of the variables from one equation and substitute it in the second equation. This way, the second equation will be

reduced to linear equation in one variable which we have already solved.

We take the second equation and find the value of as follows-

Now, by putting the value of in equation (1), we get,

Cost of 1 pen

Now, by putting the value in the equation (3) , we’ll get the value of .

Cost of 1 pencil

### Elimination Method

In this method, we eliminate one of the variables by multiplying one or both the equations by suitable non-zero constant(s) to make the coefficients of one of the variables numerically equal. Then we add or subtract one equation to or from the other so that one variable gets eliminated and we get an equation in one variable.

Since, in the above example, the numerical coefficients of one of the variables is already equal in both the equations, we will directly subtract equation (2) from equation (1) as follows

Cost of 1 pencil

Now, by putting the value of in any of the equation, we will get the value of –

Cost of 1 pen

## Graphical Method of Solving Linear Equation in Two Variables

In graphical method, we have to draw the graphs of both linear equations on the same graph sheet. The point of intersection of the two equation lines will be the solution of both simultaneous equations. The x-coordinate will give the value of x and y-coordinate will give value of y.

Example- Consider the following equations and some of the values that satisfies this equation

... (1) ... (2)

0 | 1 | -1 | |

Now by plotting these points on the same graph, we get the two equation lines and the solution of the equations. The two graphs intersect at the point whose coordinates are . Thus is the solution.