LINEAR EQUATION IN ONE VARIABLE

Linear Equations in one variable means “SINGLE VARIABLE, DEGREE ONE”. It is in the form of ax+b = 0. ( a and b are real numbers, a, b ≠ 0)

These equations help us in solving many real-life problems. We will discuss those problems further.

How To Solve Linear Equations in one variable?

Step-1: Change equation in a simple form (simplifying), if the fractional part is present in the equation, then take LCM on both sides and convert it into general form.

Step-2: Take variables including terms one side and constant terms one side. 

Step-3: Solve the equation.

You can also verify your equation by putting the solution in the equation, if you get LHS = RHS, then your answer is right, if LHS does not equal RHS, your answer is wrong, you have to check it out again.

Method for Solving Linear Equations which have one side variable and one side constant

Examples

1. Solve 9x-3=24

Sol.  9x-3 = 24

9x = 27

x = 3

1. Solve  x ⁄ 2+1 ⁄ 3= 4 ⁄ 3

Sol.  x ⁄ 2= 4 ⁄ 3 –  1 ⁄ 3

x ⁄ 2= 3 ⁄ 3 

x ⁄ 2=1

X = 2

Method for Solving those Equations having variable both side

Examples

1. 4x + 3 = 5x – 7

Sol. Firstly take all variables one side and constants another side of the equation

4x – 5x = -7 -3

-x = -10

Multiplying by minus sign both side

x = 10

1. -3x + 5 = 7x + 6

Sol. – 3x -7x = 6 – 5

  -10x = 1

-x = 1 ⁄ 10

Multiply by minus sign, bot side

x = –1 ⁄ 10

Solving Linear Equations which have to be reduced in simpler forms for getting solution of the equation

1. 3d – 2d-3 ⁄ 2 = 4 – 3(d-3)

Sol. Take LCM both sides

6d – (2d-3) = 8 – 6(d-3)

4d + 3 = 8 – 6d +18

10d + 3 = 8 + 18

10d + 3 = 26

10d = 23

d = 23/10

d = 2.3

1. (y – 4) – 2(y – 3) = 3y – 1

Sol. y -4 – 2y + 6 = 3y – 1

-y + 2 = 3y – 1

-4y = -3

  -y = -3/4

  y = 3  ⁄ 4

Some Real Life-based Problems

Let us understand real-life-based examples to better understand the topic of linear equations in one variable. 

1. The present age of Raghav’s mother is three times the present age of him. After 4 years, the age of both persons is added to 54 years, Find Raghav’s age.

Sol. For solving these types of theory questions, first, you have to suppose a variable and then write given conditions according to question, 

Let, 

Raghav’s age = x

Given, 

At present, Raghav’s mother age = 3x

After 4 years, x + 3x = 54

x + 3x = 54

4x = 54

x = 13.5

Hence, Raghav’s age is 13.5 years.

1. The ratio of the number of boys and girls in a class is 3:5. The number of boys is 6 less than girls. Find the overall strength of the class.

Sol. Let, The number of boys in the class = x

Given, No. of girls are 6 more than boys, So

Number of girls = x + 6

Ratio,

x ⁄ x+6 = 3 ⁄ 5

Now, cross multiply both sides

5x = 3 (x+6)

5x = 3x + 18

2x = 18

x = 9

Hence, Number of Boys is 9

Number of Girls is 9 + 6 = 15

Overall Strength = 24.

1. Let You and your friend play a game, your Friend asks you to think of a number in your mind. Then subtract 3 ⁄ 5 from the number and then multiply by 10. Now, He tells you that the result you get is 3 times greater than the original number. Find that original number.

Sol. Let, Original number = n

Given, 

10(n – 3 ⁄ 5) = 3n

10n – 6 = 3n

7n = 6

n = 6 ⁄ 7

Hence, The original number is 6 ⁄  7

We have done the questions above, we can also verify these answers, but how? 

So, there is a simple answer, simply put the value of the variable in the equation, then if that value satisfies the equation, then the answer is right, otherwise, check your solution once again.

Graph Of Linear Equation in One variable

If we talk about drawing a graph of a linear equation, then what will you get?

If you draw a graph of a linear equation in one variable, then you will get a straight line, it may be a horizontal or vertical line and the solution of the line will also be present on the line.

Conclusion

 As we were talking about the topic of linear equations in one variable, it is widely used for finding solutions to the equations. A lot of daily life problems can be solved by using this technique. The only thing to remember is that the power of the variable present in the equation should be one only. Above we have discussed the step-by-step ways of solving the linear equation in one variable. We have also talked about ways of solving the equations which have one side variable and having both side variables too. We have discussed the plotting of graphs of linear equations in one variable.