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Linear Equation

Introduction to Linear Equations:

Look at these statements.

(i) 8 + 5 = 13; (ii) 28 - 7 = 21

These statements have the symbol '=', called the equality sign. We can state whether these statements are true or false. Thus, a mathematical statement with an equality sign is called a statement of equality.

Now look at these statements:

(i) p + 3 = 15

(ii) z - 5 = 8

(iii) 2m = 6

(iv) x + 3 = 14

Such a statement is called an equation. Thus, a statement of equality which involves one or more variables is called an equation.

Some statements along with their equations are:

Statements

Equations

(1) A number x increased by 8 is 24.

(ii) 7 exceeds a number z by 4.

(iii) 8 times a number x is 40.

(iv) A number k divided by 4 is 7.

x + 8 = 24

7 - z = 4

8x = 40

$\frac{k}{4}$ = 7

Also note that the power of the variable in each equation is only one. Such equations are called linear equations in one variable.

What is a linear equation?

An equation which involves only one variable whose highest power is 1 is known as a linear equation in that variable.

For example:

(a) x + 4 = 19

(b) y - 7 = 11

(d) 2x - 5 = x + 7

(e) x + 13 = 27

(f) y - 3 = 9

(g) 11x + 5 = x + 7

Each one of these equations is a linear equation.

Solution of an Equation:

The sign of equality divides the equation into two sides. Left hand side or L.H.S. and Right hand side or R.H.S

Solution of linear equation or Root of linear equation: The value of the variable which makes left hand side equal to right hand side in the given equation is called the solution or the root of the equation.

Definition:

The value of the variable for which a statement is true, is called the solution or root of the equation.

For example:

1. x + 1 = 4

Here, L.H.S. is x + 1 and R.H.S. is 4

If we put x = 3, then L.H.S. is 3 + 1 which is equal to R.H.S.

Thus, the solution of the given linear equation is x = 3

2. 5x - 2 = 3x - 4 is a linear equation.

If we put x = -1, then L.H.S. is (5 × - 1) - 2 and R.H.S. is (3 × - 1) - 4

= -5 -2 = -3 -4

= -7 = -7

So, L.H.S. = R.H.S.

Therefore, x = -1 is the solution for the equation 5x - 2 = 3x - 4

Note: The sign of equality in an equation divides it into two sides, namely LHS (Left Hand Side) and RHS (Right Hand Side). The value of LHS is equal to the value of RHS. If the LHS is not equal to the RHS, we do not get an equation. For example, (x - 4) > 8 or (x - 4) < 8 are not equations.

Solving an Equation by Trial and Error Method:

A simple way of finding the solution of an equation is to give several values for the variable, say x, and find LHS and RHS. When LHS = RHS for a particular value of the variable, we say that it is a root of the equation. This method is known as trial and error method.

Let us consider the following statements:

p + 2 = 8 ..... (i); p - 2 = 9 ..... (ii); $\frac{p}{5}$ = 12 ..... (iii)

Finding the value of the variable for which the given statement is true, is called solving an equation.

Statement (i) is true only when p = 6

Statement (ii) is true only when p = 11.

Statement (iii) is true only when p = 60.

For any other value of p, these statements are not true.

Examples on Solving an Equation by Trial and Error Method:

1. Solve: 3x - 5 = 4

Solution:

We try several values of x to find the LHS and RHS.

When x = 1, LHS is which is 3 × 1 - 5 = - 2 ≠ RHS, which is 4.

When x = 2, LHS is which is 3 × 2 - 5 = 1 ≠ RHS, which is 4.

When x = 3, LHS is which is 3 × 3 - 5 = 4 = RHS, which is 4.

Hence, x = 3 is the solution of the given equation.

2. Using trial and error method, find the solution of the equation 5x = 20

Solution:

We try several values of x to find the LHS and RHS.

When x = 1, LHS is 5 × 1 = 5 ≠ RHS, which is 20.

When x = 2, LHS is 5 × 2 = 10 ≠ RHS, which is 20.

When x = 3, LHS is 5 × 3 = 15 ≠ RHS, which is 20.

When x = 4, LHS is 5 × 4 = 20 = RHS, which is 20.

Hence, x = 4 is the solution of the given equation.

3: Using trial and error method, find the solution of the equation $\frac{3}{4}$ x + 4 = 7

Solution:

We try several values of x to find the LHS and RHS.

When x = 1, LHS = $\frac{3}{4}$ × 1 + 4 = $\frac{3}{4}$ + 4 ≠ RHS, which is 7.

When x = 2, LHS = $\frac{3}{4}$ × 2 + 4 = $\frac{3}{2}$ + 4 ≠ RHS, which is 7.

When x = 3, LHS = $\frac{3}{4}$ × 3 + 4 = $\frac{9}{4}$ + 4 ≠ RHS, which is 7.

When x = 4, LHS = $\frac{3}{4}$ × 4 + 4 = 3 + 4 = 7 = RHS, which is 7.

Hence, x = 4 is the solution of the given equation.

How to solve linear equation in one variable?

Rules for solving a linear equation in one variable:

The equation remains unchanged if –

(a)The same number is added to both sides of the equation.

For example:

1. x - 4 = 7

⇒ x - 4 + 4 = 7 + 4 (Add 4 to both sides)

⇒ x = 11

2. x - 2 = 10

⇒ x - 2 + 2 = 10 + 2 (Add 2 to both sides)

⇒ x = 12

(b) The same number is subtracted from both sides of the equation.

For example:

1. x + 5 = 9

⇒ x + 5 - 5 = 9 - 5 (Subtract 5 from both sides)

⇒ x + 0 = 4

⇒ x = 4

2. x + 1/2 = 3

x + 1/2 - 1/2 = 3 - 1/2 (Subtract 1/2 from both sides)

⇒ x = 3 - 1/2

⇒ x = (6 - 1)/2

⇒ x = 5/2

⇒ x/2 × 2 = 5 × 2 (Multiply 2 to both the sides)

⇒ x = 10

2. x/5 = 15

⇒ x/5 × 5 = 15/5 (Multiply 5 to both the sides)

⇒ x = 3

(d) The same non-zero number divides both sides of the equation.

For example:

1. 0.2x = 0.24

⇒ 0.2x/0.2 = 0.24/0.2 (Divide both sides by 0.2)

⇒ x = 0.12

2. 5x = 10

⇒ 5x/5 = 10/5 (Divide both sides by 2)

⇒ x = 2

What is transposition? Explain the methods of transposition.

Any term of an equation may be shifted to the other side with a change in its sign without affecting the equality. This process is called transposition.

So, by transposing a term —

● We simply change its sign and carry it to the other side of the equation.

● ‘+‘ sign of the term changes to ‘—‘ sign to the other side and vice-versa.

● ‘×’ sign of the factor changes to ‘÷‘ sign to the other side and vice-versa.

● Now, simplify L.H.S. such that each side contains just one term.

● Finally, simplify the equation to get the value of the variable.

For example:

10x - 7 = 8x + 13