A linear equation will not involve any products and roots of variables and all these variables occur only to the first power and do not appear for the assessment of logarithmic, trigonometric, or even exponential functions.
Few examples of linear equations are given below:
$3x + 5y = 12$, $\frac{1}{2}$$x + 7y + 2z = -9$, $x_{1} - 2x_{2} - 3x_{3} + x_{4} = 0$
Linear equation with 2 variables is very easy and interesting topic. Sometimes when students get stuck with any math problem and might think whether to continue that question or not? This makes difficult to solve problems, as well as making it difficult to easily absorb new ideas or information. Math is all about practice. More you practice more you get confidence. Students are always advice to understand the concept before they attempt any math problem on any topic. In this section, Class 9 students will get step-by-step explanation to every question specified in the linear equation chapter. Because jumping directly into solving problems can lead to frustration and confusion.

General Form

A linear system of two equations with two variables in any system is written as

ax + by = p
cx + dy = q
Where: x and y are variables
a, b, c, and d are coefficient of x and y and should not be negative.
p and q are constant values. Can be any real number.

Methods to Solve Linear Equations in Two Variables

Many systems in all kinds of diverse fields can be characterized by the input-output analysis. Generally, to analyse such systems, we follow three different methods:

Method 1: The system is described by some specific mathematical form, in which the input is given and an output is to be found.
Method 2: The system and the output are provided and we need to find the input.
Method 3: Both the input and the output are provided and we need to create the system.

This system of finding the solution from the given input and output gives us the linear system or equations. The general method for solving a linear equation is to perform algebraic operations on the system that will not alter the solution set and will produce a successive simpler system, until a point is reached where it can be checked whether the system is consistent or not and what the solutions are.

A finite set of linear equations are called a system of linear equations or more accurately a linear system. The variables in these are unknowns.
The algebraic operations will follow as mentioned:
1. Multiply an equation by a non-zero constant
2. Interchange the two equations
3. Add constant times of one equation to another

Examples

Example 1: By using the method of addition, solve

3x  + 5y  =  8
8x  -  5y  =  12

Solution:
Consider the given equations
3x  + 5y  =  8
8x  -  5y  =  12

Add the given equations
3x  + 5y  =  8                                       
8x  - 5y  =  12
__________________
11x + 0y = 20
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$\Rightarrow$  x = $\frac{20}{11}$

Substitute x = $\frac{20}{11}$ in   3x  + 5y  =  8

We get, 3($\frac{20}{11}$)  + 5y  =  8

Solve for y

$\Rightarrow$  5y = 8 - ($\frac{60}{11}$)

55y = 88 - 60

y = ($\frac{28}{55}$)

Therefore, x= $\frac{20}{11}$ and y = ($\frac{28}{55}$)

Example 2: Using substitution method solve :  x + y = 9 and x - y  = 3
Solution: 
Isolate x from x + y = 9 
$\Rightarrow$  x = 9 -  y
Substitute x = 9 -  y in x - y  = 3
We get,
9 - y - y = 3
9 - 2y = 3
-2y = -6
$\Rightarrow$ y = 3

Now you can substitute y = 3 in any one of the given equations to find the value of x

In x + y = 9  substitute y = 3 and solve for x
x + 3 = 9
$\Rightarrow$ x = 6

Therefore (x, y) = (6, 3).

Practice Questions

Practice below problems.
Question 1: Solve for x and y
3x - 2y = 1 and x - y = 10 
Question 2: Solve below given system using substitution method.
2x - y = 3 and
x + y = 5