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Linear equation in One variable



What is algebraic expression

Algebraic expression is the expression having constants and variable. It can have multiple variable and multiple power of the variable
Example
10x
2x -3
3x + y
2xy + 11
x2 +2
 

What is algebraic equation

We already know the below terms from previous class
What is equation
An equation is a condition on a variable.
What is variable
A variable takes on different numerical values; its value is not fixed. Variables are denoted usually by letters of the alphabets, such as x, y, z, l, m, n, p, etc
 
An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
Example
5x = 25
2x – 3 = 9,
11xy+1 =98
x2 +1=y2
Important points to remember
1)These all above equation contains the equality (=) sign.
2) The above equation can have more than one variable
3) The above equation can have highest power of variable > 1
4) The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS)
5) In an equation the values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variable. These values are the solutions of the equation.
6) We assume that the two sides of the equation are balanced. We perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. We get the solution after generally performing few steps
 

What is Linear equation in one Variable

We will restrict the above equation with two conditions
a) algebraic equation in one variable

b) variable will have power 1 only
Example
5x = 25
2x – 3 = 9
We will focus on Linear equation in One variable in this chapter.

Solving Equations which have Linear Expressions on one Side and Numbers on the other Side

2x -3= 5
3x-11= 22
How to solve them
1) Transpose (changing the side of the number)  the Numbers to the side where all number are present.  We know the sign of the number changes when we transpose it to other side
2)  Now you will have a equation have variable on one side and number on other side. Add/subtract on  both the side to get single term
3) Now divide or multiply on both the side to get the value of the variable
Example
2x -3= 5
Solution
Transposing 3 to other side
2x=5+3
2x=8
Dividing both the sides by 2
x=4
Watch this tutorial on how to solve Equations which have Linear Expressions on one Side and Numbers on the other Side

Word Problem on Simple Linear equation in one variable

Linear equation can be used to solve many word Problem. The procedure is simple
1) First read the problem carefully. Write down the unknown and known
2) Assume one of the unknown to x  and find the other unknown in term of that
3) Create the linear equation based on the condition given
4) Solve them by using the above method
Example:
The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool?
Solution
 The unknown are length and breadth.
Let the breadth be x m.
Then as per question the length will be (2x + 2) m.
Perimeter of swimming pool = 2(l + b) = 154 m
2(2x + 2 + x) = 154
2(3x + 2) = 154
Dividing both sides by 2
3x + 2 = 77
Transposing 2 to R.H.S, we obtain
3x = 77 − 2
3x = 75
Dividing 3 on both the sides
x = 25
So
Breath is 25 m
Length =2x + 2 = 2 × 25 + 2 = 52m
Hence, the breadth and length of the pool are 25 m and 52 m respectively.
Watch this tutorial on Word Problem on Simple Linear equation in one variable

 

Solving Equations having the Variable on both Sides


2x -3= 6-x
3x-11= 4x
How to solve them
1) Here we Transpose (changing the side of the number) both the variable and  Numbers to the side so that one side contains only the number and other side contains only the variable. We know the sign of the number changes when we transpose it to other side.Same is the case with Variable
2)  Now you will have a equation have variable on one side and number on other side. Add/subtract on  both the side to get single term
3) Now divide or multiply on both the side to get the value of the variable
Example
2x -3= 6-x
Solution
Transposing 3 to RHS and x to LHS
2x+x=6+3
3x=9
Dividing both the sides by 3
x=3
Watch this tutorial on Solving Equations having the Variable on both Sides

 


Reducing Equations to Simpler Form

You will find many situations where the linear equation may be having number in denominator. We can perform the below steps to simplify them and solve it
1) Take the LCM of the denominator of both the LHS and RHS
2) Multiple the LCM on both the sides, this will reduce the number without denominator and we can solve using the method described above
Example
Solve the linear equation 
Linear equation in one variable
Solution
Linear equation in one variable
L.C.M. of the denominators, 2, 3, 4, and 5, is 60.
So Multiplying both sides by 60, we obtain
 30− 12 = 20+ 15 +60
Transposing 20x  to R.H.S and 12 to L.H.S, we obtain
30− 20x = 15 + 12+60
10x = 87
Dividing both the sides by 10
x=87/10=8.7 
Watch this tutorial on Reducing Equations to Simpler Form

Equations Reducible to the Linear Form

Let’s explain this with an example
The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.
Solution
 Let the common ratio between their ages be x. Therefore, Hari’s age and Harry’s age will be 5x years and 7x years respectively and four years later, their ages will be (5x + 4) years and (7x + 4) years respectively.
According to the situation given in the question,
Linear equation in one variable
Now this is not a linear equation as we studied, but if you observe, we can reduce to linear equation.
We multiply both the sides by (7x+4)
Multiplying both the sides by (7x+4)
(5x+4) =3(7x+4)/4
or
20x+16=21x+12
Transposing 20x to RHS and 12 to LHS
4=x
x=4
  Hari’s age = 5x years = (5 × 4) years = 20 years
Harry’s age = 7x years = (7 × 4) years = 28 years
Therefore, Hari’s age and Harry’s age are 20 years and 28 years respectively.
 
 
Watch this tutorial on how to solve word problem in linear equation


 

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