**model**real life situations. For example: we burn approximately 118 calories per hour of walking. How long should we walk to burn 354 calories ? To determine this, we use equation:

**118t = 354**, where 't' is the time. The answer is 3 hours.

## Linear Equation in One Variable

**Equation: **An equation is a statement in which two algebraic expressions are equal. In other words, an equation is a sentence that expresses the equality of two algebraic expressions. Ex: 4x – 3y = 2; x^{2} + y^{2} + z^{2} = 10; p^{2} + 17 = 2p^{2} + 4q^{2}. In the first example 4x – 3y = 2, 4x – 3y is called left hand side (L.H.S.) of the equation and 2 is called right hand side (R.H.S.) of the equation.

**Linear equation in one variable:** An equation which involves only one variable with exponent 1, is called a linear equation in that variable. The general form of a linear equation in one variable is: **ax + b = 0**

where 'a', 'b' are real numbers, ‘x’ is a variable and a ≠ 0.

Each one of the equations: 3x + 2 = 0; 4p – 5 = p; 2y + 3 = 3y + 4 is a linear equation in one variable.

A value of the variable which when substituted for the variable in the equation, makes its two sides (LHS and RHS) equal, is called a **solution** (or **root**) of the equation.

To solve a linear equation in one variable, we have to find the value of the variable satisfying the given equation. Consider the equation 3x + 2 = 8. Because 3(2) + 2 = 8 is true, we say that 2 satisfies the equation. Therefore, 2 is the solution or root for given linear equation.

**Rules for solving a linear equation: **

The equality of a linear equation is not changed, when:

(i) the same number is added to both sides of the equation

(ii) the same number is subtracted from both sides of the equation

(iii) both sides of the equation are multiplied by the same non-zero number

(iv) both sides of the equation are divided by the same non-zero number.

**Ex:**Determine whether ‘5’ is a solution of the linear equation 4x – 3 = 3x + 2.

**Sol:**Substitute 5 for 'x' and evaluate each side of the equation. Given linear equation is: 4x – 3 = 3x + 2. By substituting x = 5, we get: L.H.S. = 17 and R.H.S. = 17. The two sides of the equation have same values. So, 5 is the solution for the given equation.

**Transposition: **Any term of an equation may be taken to the other side with its sign changed i.e, + becomes – and – becomes +. It does not affect the equality. This process is called transposition.

For example: 4x – 2 = 2x – 6 4x – 2x = – 6 + 2.

Here, the term involving ‘x’ in R.H.S. has been transposed to L.H.S. So 2x in R.H.S. becomes – 2x in L.H.S. The constant term (– 2) from L.H.S. has been transposed to R.H.S. as + 2.

Adding the like terms, we have 2x = – 4.

On transposing the constant term (– 4), we have 2x + 4 = 0.

**Cross-multiplication: ** If **=** , then **(a × d) = (b × c)**, that is, **ad = bc**.

The numerator on L.H.S. is multiplied with denominator on R.H.S. and numerator on R.H.S. is multiplied with denominator on L.H.S., with "=" sign in tact.

This process is called cross-multiplication.

Ex: Consider the equation .

Then, by cross-multiplication, we get: 2(2x + 3) = 5(x – 4).

The method for solving a system of linear inequalities in two variables graphically is explained through an example.

**Example:** Solve the following system of linear inequalities graphically:

x + y ≥ 5 ----- (i)

x – y ≤ 3 ----- (ii)**Sol:** Let us consider the linear equality x + y = 5

Its graph is shown in figure.

The solution of inequality (i) is represented by the shaded region above the line x + y = 5, including the points on the line.

Now consider the linear equality x – y = 3

Its graph is shown in figure.

The solution of inequality (ii) is represented by the shaded region above the line x – y = 3, including the points on the line.

The region common to the above two linear inequalities is the required solution. It is represented by the darker shaded region in the adjacent figure.

**The relationship between two or more variables is called a formula.**

(The plural of a formula is formulae).

Ex: v = u + at gives the relationship between the variables 'v' (final velocity), 'u' (initial velocity), 'a' (acceleration) and 't' (time). Similarly, h^{2} = a^{2} + b^{2} is a formula relating the hypotenuse (h) of a right-angled triangle to its two sides (a and b).

** Framing a formula: ** To frame a formula, we have to obtain the relation between given variables based on the conditions stated.

Ex: The perimeter of a rectangle is twice the sum of its length and breadth. So, if 'P' is the perimeter, 'l' is the length and 'b' is the breadth of the rectangle, we have P = 2(l + b).

**Derived or auxiliary formulae: **If I = , then P = ; T = ; R = .

These new formulae deduced from original formula I = are called 'derived' or 'auxiliary' formulae.

**Definition:** The subject of a formula is the variable which is expressed in terms of the other variables.

Consider the formula for volume of a cylinder, ie. V = πr^{2}h. In this formula, the subject is 'V' (volume), because it is expressed in terms of other variables 'r' (radius) and 'h' (height).

**Change of subject: ** In a given formula, we may express any of the variables in terms of the other variables by using the rules applicable in solving an equation. Thus, in a given formula, we may change the subject to the desired one.

Characteristics of subject in a formula |
---|

• The subject symbol occurs on the L.H.S. of the equality sign. |

• It is written independently without being linked with any other quantities or variables. |

• Its coefficient is always one. |

• All properties used in simple equations are also used in transforming the subject of a formula. |

The formula for volume of a cylinder with its height (h) as the subject is

Similarly, the formula for volume of a cylinder with its radius (r) as the subject is