In this chapter, we will learn the concept of linear equation, its properties and solved examples.
What are Linear Equation?
Linear equation can be understood with following points;
(a) It’s an equation.
This means that the math expression contains ” = ” symbol in it.
(b) The highest power of variable in the given term is 1.
In linear equations we have math terms made of constants & variables with power 1.
Examples of Linear Equation
(i) 7x + 2 = 6
It’s an example of linear equation since the highest power of variable x is 1.
(ii) 5x + 3y = 4
In the above equation we have two variable terms 5x & 3y. Both the terms have power 1, hence the equation is linear equation.
(iii) 8xy + 2y + 3 = 0
Consider the terms of above equation;
8xy ⟹ two variables x & y are multiplied together. Hence the power of term 8xy is 2.
2y ⟹ Highest power is 1
3 ⟹ Power is 0
Since the equation’s highest power is 2, it is not a linear equation.
(iv) 9x + 6y + 2z + 9 = 0
Consider the terms of above equation.
9x ⟹ power is 1
6y ⟹ power is 1
2z ⟹ power is 1
9 ⟹ power is 0
Since the highest power of equation is 1, the above equation is linear equation.
(v) \mathtt{9x^{2} +2xy^{2} +3y\ =\ 24}
Consider all the terms of given equation.
\mathtt{9x^{2}} ⟹ Power is 2
\mathtt{2xy^{2}} ⟹ Power is 3
3y ⟹ Power is 1
24 ⟹ Power is 0
Since the highest power of given equation is 3, the equation is not linear equation.
Solving Linear Equation
Solving Linear equation means finding the value of variables in given equation.
Some important point to note;
(a) If there is only one variable in linear equation, then we need only one equation to find value of the variable.
(b) If there are two variables present in linear equation, then we need to have two equations to find value of those two variables.
Hence, if there are “n” number of variables in linear equation, then we need to have “n” number of different equation to find value of all the variables.
In this chapter we will stick to solving equations with one variable.
Solving Linear Equation with one variable
Here we will learn to solve linear equation using transposition method.
In transposition method, when we move any term from one side of equation to other side, following changes occur;
(a) Addition change into subtraction and vice-versa.
(b) Division change into multiplication and vice versa.
Let us solve some examples for our understanding.
Example 01
Solve 6y + 4 = 22
Solution
The above equation is linear equation since the highest power of variable is 1.
Solving the equation using transposition method.
6y + 4 = 22
Take number 4 to the right.
While moving number the addition will change into subtraction.
6y = 22 – 4
6y = 18
Now take number 6 to right.
Here multiplication will change into division.
6y = 18
y = 18 / 6
y = 3
Hence for the given linear equation, the value of y is 3.
Example 02
Solve \mathtt{\frac{x}{2} \ -\ 5\ =\ 13}
Solution
The given equation is linear equation since the highest power of any given term is 1.
Solving the equation using transposition method.
Moving number 5 to right side.
Here the subtraction becomes addition.
\mathtt{\frac{x}{2} \ -\ 5\ =\ 13}\\\ \\ \mathtt{\frac{x}{2} \ =\ 13\ +\ 5}\\\ \\ \mathtt{\frac{x}{2} =\ 18}
Move number 2 to the right side.
Division becomes multiplication.
\mathtt{\frac{x}{2} =\ 18}\\\ \\ \mathtt{x\ =\ 18\ .\ 2}\\\ \\ \mathtt{x\ =\ 36}
Hence, the value of x is 36.
Example 03
Find value of x & y in given equation.
x + 9y = 12
Solution
Here the linear equation contains two variables.
In order to find value of two variable using transposition method, we need at least one more linear equation.
Hence, we cannot find the value of x & y using transposition method.
However, you can find value of x & y using hit & trial method.
Just randomly put the value of x & y and check if it satisfies the given equation.
Put x = 1 & y =1 in equation.
x + 9y = 12
1 + 9 = 12
10 = 12
The equation is not satisfied.
Put x = 3 & y = 1
x + 9y = 12
3 + 9 = 12
12 = 12
The equation is satisfied, hence x = 3 & y = 1 is the solution.
Hit & Trial method is very time consuming technique. That’s why we try to avoid using the method for solving linear equations.
Graph of Linear Equation
The graph of linear equation is always a straight line.
Here we will look at the graph of;
(a) Linear equation with one variable
(b) Linear equation with two variable
Graph of linear equation with one variable
Consider the linear equation;
x + 4 = 6
Given below is the graph of linear equation.
The red line parallel to y axis is the graph of x + 4 = 6.
Hence, the graph of linear equation with one variable is always straight line parallel to x or y axis depending upon the equation.
Graph of Linear equation with two variables
Let us consider below equation;
x + 2y = 7
The red line with some slope is the graph of given equation.
Hence, the graph of linear equation with two variables is always straight line with some slope depending upon the given equation.