There are many types of equations. Each equation represents a specific type of graph on a numbered grid. Today we’re going to learn a few things about linear equations. First let us understand why linear equations are called ‘linear equations’? An equation that when plotted yields a straight line is called a linear equation. That’s confusing, right? Don’t worry. We will look at this in more detail in just a bit. NCERT Solutions on chapter linear equations in 2 variables for class 10 are the most important and foremost tool to look up to, especially when students are preparing for school and competitive examinations. Students get complete and tailored answers for each questions. We also provides step by step solutions and well-illustrated examples on linear equations in 2 variables.

What is Linear Equations in Two Variables

An algebraic equation with two variables where degree of each variable is one. The general form of the linear equations system having variables x and y is :
$a_1$x + $b_1$y = $c_1$
$a_2$x + $b_2$y = $c_1$
Where $a_1$, $b_1$, $a_2$ and  $b_2$ not be equal to zero.

There are many ways to solve the simultaneous set of two linear equations the common method are 

1) Solve by Graphing

2) Solve by Elimination (also called as by Addition)

3) Solve by Substitution

4) Solve by using Matrix

Methods to Solve Linear Equation in Two Variables

Class 10 students should be get handy with the below listed methods to solve linear equations with 2 variables. These can also prove to be of valuable help to students in their assignments and preparation of boards and competitive exams.

Solving Systems of Linear Equations by Substitution


Solve one of the equations with one of the variables and substitute in the remaining equations which gives one equation with one variable which can easily be solved. Once solved substitute the value back in one of the given equations and find the value of the remaining variables.

Number of equations should be equal to number of variables.

Systems of Linear Equations by Elimination

Elimination method is also known as Addition or Subtraction method. Transform the variables in such a way that one variable cancels out when solved simultaneously.

Solving Systems of Linear Equations by Graphing


In graphical method there can be one, none or infinitely many solutions. If for the given system of equations, we can graph a straight line then it possible to solve them graphically.
Draw graphs for both the lines and record their point of intersection, which is the solution.

Simultaneous Linear Equations in Two Variables

Simultaneous linear equations with two variables $x$ and $y$ would have a set of $2$ equations and we have $t$ solve for the point of intersection $(x, y)$.  The solution to the set may be consistent or inconsistent. 

Consistent: When the simultaneous linear equation has a solution or infinite numbers of solutions then the equations are said to be consistent.

The set having a single solution is called Independent.  

The set of simultaneous equations having infinite solutions are called dependent.

Inconsistent: When the set of simultaneous equations don't have any solutions then it is called inconsistent. 

To understand about solution to simultaneous set of linear equation in two variables we need to understand, if we graph them on a plane and they can meet at one point (one solution, consistent and independent). If the lines are one and same and meet at all points (multiple solutions so it’s called consistent and dependent). If the lines are parallel (no solution, inconsistent)

Example of a set of simultaneous linear equation

$2x + 3y$ = $10$

$3x + y$ = $25$

Examples

Illustrated some examples on pair of linear equations in two variables class 10 students below:
Example 1: Solve by elimination method:
  x  - 7y  = 9
-2x + 3y  = 3

Solution: The given system of equations are:
  x  - 7y  = 9                      --------->   1
-2x + 3y  = 3                      --------->   2

Multiply the first equation by 2.
2x - 14y = 18                        --------->  3

Solve third and second equation simultaneously.
2x - 14y = 18    
-2x + 3y  = 3      
_______________
  0x    -11y  = 21
_______________

$\Rightarrow$ y  = $\frac{-21}{11}$

Substitute y = $\frac{-21}{11}$ in the first equation and solve for y

x  -  7 ($\frac{-21}{11}$) = 9
$\Rightarrow$  x + $\frac{147}{11}$  = 9
$\Rightarrow$   11x = 99 - 147
 
$\Rightarrow$   x = $\frac{-48}{11}$ 

Therefore x = $\frac{-48}{11}$  and y = $\frac{-21}{11}$

Verification can be done by substituting  x and y values in one of the given equations.

Example 2: Solve the pair of linear equations by substitution.
3x + y = 5 and
2x - y = 0

Solution:

3x + y = 5 ----(1)

2x - y = 0 ---- (2)

Now we can write any one of the equations in term of $y$.

2x - y = 0

Add y to both sides we get 2x = y we can say y = 2x

Now substitute y = 2x in (1) i.e. replace y with 2x.

We get 

3x + (2x) = 5

5x = 5

x = 1

We know y = 2x i.e y = 2(1) = 2

The solution is (1, 2).

Practice Problems

Solve below problems.
Problem 1: Solve 8x - 6y = -20
16x - 7y = -30
Problem 2: Solve for x and y:
x + 2y = 8
2x + y = 15