Math's: Linear Equations: Formation and Solutions of Linear Equations in One Variable

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Linear Equations

An expression that contains an ‘=‘ sign is an equation. An equation that contains a polynomial(s) of degree one is known as Linear equation. Example-

Linear Equations

Linear Equations

  • A Linear equation that contains only one variable is known as linear equation in one variable. It can be written in the form of where and are real numbers and is the variable.

  • An equation which contains two variables and the exponents of each variable is one and has no term involving product of variables is called a linear equation in two variables. For Example, . The general form of a linear equation in two variables is where andare real numbers such that at atleast one of and is non-zero and and are variables.

1. State which of the following is/are linear equations.

(i)

(ii)

(iii)

(iv)

Sol- (i) contains polynomial of degree one. It is a linear equation.

(ii) contains a polynomial of degree one. It is a linear equation.

(iii) contains a polynomial of degree 3. It is not a linear equation.

(iv) contains a polynomial of degree 2. It is not a linear equation.

2. State which of the following is/are linear equations in one variable and which is/are linear equations in two variables.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Sol- (i) is a linear equation in two variables as there are two variables and .

(ii) is a linear equation in one variable since it has only one variable .

(iii) is a linear equation in two variables as it has two variables and .

(iv) is a linear equation in one variable as there is one variable .

(v) is not a linear equation since it includes the polynomial of degree 2.

(vi) is a linear equation in two variables as there are two variables and .

Formation and Solution of Linear Equations in One Variable

Example- Thrice a number added to 12 gives 36. Find the number.

Formation of Linear Equation- Here, the number is unknown, so it can be assumed as variable .

Then, thrice the number will be

Adding to gives . So, this relationship can be shown in the form of equation as-

Solution of Linear Equation

The linear equation formed will be solved as follows:

  • Transpose the constant term at one side (RHS) so that the term with variable remains at one side (LHS) while all the constant is one the other side (RHS). When a term is transposed from one side to other, sign changes to and changes to .

  • Now, the given equation will be changed as

  • The constants and the variables are then added or subtracted on both the sides. i.e.

  • The numerical coefficient with the variable will be transposed to the other side such that the numerator of the coefficient becomes the denominator of the number on the other side and vice versa.

  • So, the required number is .

3. The sum of three consecutive odd numbers is 33. Find the numbers.

Solution: Let the smallest number be ,

Then, the next consecutive odd number will be

The third next consecutive odd number will be

As per the question, the linear equation will be

  • The smallest number

  • The second consecutive odd number

  • The third consecutive odd number

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