Maths Notes on Linear Equations

By Rahul Chadha|Updated : December 7th, 2018

What is a linear equation? 

Any equation of the first degree is known as a linear equation.

Linear equation in one variable - It is an equation of the form ax + b = 0, a 0, where a and b are constants and x is a variable.

ax + b = 0
byjusexamprep

byjusexamprep is known as the solution or root of the equation ax + b = 0.

Example

Consider the following equations: 

1)2x + 1 = 0, 

byjusexamprep

Solution

The root of the equation is x = -byjusexamprep

 

byjusexamprep

 

Example

3x - 2 = 0,

byjusexamprep

Solution

The equation is x = byjusexamprepor it is called the root of the equation.

 

byjusexamprep

 

Example

4x +7= 3

Solution

The solution of this equation is 4x = 3 – 7

x = byjusexamprep

byjusexamprep is the root of the equation.

POINTS TO NOTE

The solution of a linear equation is not affected when

1)the same number is either added or subtracted from both sides of the equation.

2) the same non-zero number is multiplied or divided on both sides of the equation.

Linear Equation in Two Variables

It is an equation of the form ax + by + c = 0, where a and b are real numbers, a ¹ 0, b ¹ 0 and c are constants (real numbers) and x, y are variables. Let us look at the following linear equation in 2 variables

 1) x + 2y = 3,   

Here x + 2y = 3 can be written as 

 x + 2y - 3 = 0. Here it is in the form ax + by + c = 0 where a =1, b =2 and c = -3

 
2) 2x - 3y - 5 = 0

It is in the form ax + by + c = 0 where a =2, b =-3 and c = -5


3)  5x+7y = 4,

Here 5x + 7y = 4 can be written as

5 x + 7y - 4 = 0

It is in the form ax + by + c = 0 where a =5, b =7and c = -4


Solution 
A pair of values of x and y satisfying the equation ax + by + c = 0 is known as the solution of the equation.

Example : x = 1, y = 2 is a solution of x + 3y = 7.

Solution of a Linear Equation

We observed that every linear equation in one variable has a unique solution. The solution of a linear equation in two variables is a pair of values (x, y) which satisfy the given equation. Consider an equation 2x + 5y = 17. Here, x = 1 and y = 3 is a solution as when we substitute x = 1 and y = 3 in the equation, we get2x + 5y = 2´1 + 5´ 3 = 2 + 15 = 17

Thus the solution is (1, 3). Similarly (6, 1) is also a solution for the above equation. Try the pair (0, 3). L.H.S = 2´0 + 5´3 = 0 + 15 = 15, which is not 17. Hence (0, 3) is not a solution to the above equation. But this doesn’t mean that there are no more solutions. We can get many solutions in the following way: 

Example

Let x = 3 and 2 ´ 3 + 5y = 17 i.e., 5y = 17 – 6 Þ 5y = 11 Þ y = byjusexamprep.

Solution

byjusexamprep.


So there is no end to different solutions of a linear equation in two variables. That is, a linear equation in two variables has infinitely many solutions.

Example :


1) Find four solutions of the equation x + 3y = 7

Let us choose x = 1 then 3y =6 and hence y =2

Therefore x = 1 and y =2 is a solution of the equation x + 3y = 7

If we take x =4 then 3y =3 or y =1

Therefore x = 4 and y =1 is a solution of the equation x + 3y = 7

If we assume x =0 then y = byjusexamprep,is a solution of the equation x + 3y = 7

If we assume y = 0 then x = 7, is a solution of the equation x + 3y = 7

(1, 2), (4, 1), (0, byjusexamprep), (0 ,7) are the four solutions of the infinitely many solutions.

2) Find two solutions of the equation 5x + 3y = 8

Let us assume x=1 then 3y =8 - 5 =3y=3Þ byjusexamprep.

 x=1 and y = 1 is a solution of the equation 5x + 3y = 8

Let us assume x = 0 then y = byjusexamprep

x = 0 and y = byjusexamprepis a solution of the equation 5x + 3y = 8.

Hence (1, 1) and (0, byjusexamprep) are the two solutions of the infinitely many solutions.

Graph of Linear Equation in Two Variables

When we draw the graph of the equation ax + by + c = 0, we get a straight line.

Note: The graph of a first degree equation in one or two variables is a (straight) line. Because of this reason we call it a linear equation.


byjusexamprep

Draw the graph of y = 2x - 1. Use it to find some solutions of the equation. Check from the graph whether x = -1, y = -3 is the solution.

y = 2x – 1
If x = 0; y = 2 × 0 - 1 = 0 - 1= -1
If x = 1; y = 2 × 1 - 1 = 2 - 1 = 1
If x = 2; y = 2 × 2 – 1 = 4 - 1 = 3

We thus have the following points:


byjusexamprep

 

x

0

1

2

y

-1

1

3


Plotting A (0, -1), B (1, 1), C (2, 3), we draw the graph.

We see that the point P (-1, -3) lies on the line, which shows that x = -1 and y = -3 is a solution of the equation y = 2x - 1. Q(byjusexamprep, -4) and R (-2, -5) lie on the graph indicating that

x =byjusexamprep, y = -4 and x = -2, y = -5 are some other solutions of the equation.

 Equations of Lines Parallel to the x-axis and y-axis

We would observe that all the points with y-co-ordinate 0 lie on the x-axis and that with x-co-ordinate 0 lies on the y-axis. 
Now we can guess the equation of the line parallel to the x-axis as y = k and y-axis as x = k. Let us draw the graph of the equation x – 2 = 0. According to a linear equation in one variable this equation has one solution x = 2. 
Considering it as an equation in two variables, it can be expressed as x + y.0 = 2. This equation has infinitely many solutions as y can take any real number.

 

byjusexamprep


Let us draw the graph of the equation y – 3 = 0. According to a linear equation in one variable this equation has one solution y =3.

Considering it as an equation in two variables, it can be expressed as 0.x + y = 3. This equation has infinitely many solutions as x can take any real number.

 

byjusexamprep

Summary

 

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