A linear equation is basically an equation consisting of terms where each term can be either a single variable and constant or product of a constant and a variable. It is an algebraic equation which may have more than one variable.

A linear equation with two variables is of the form:

Y = mX + c; here in this equation ‘Y’ and ‘X’ are two variables and ‘m’ (coefficient of ‘X’) is a Slope or gradient which is a constant. Also ‘c’ is a constant which indicates the Point where line touches the y- axis of the plane i.e. ‘Y’ intercepts.

Some of the examples of the linear equation is:

1. Y= mX + c.

It is a simple equation of line where ‘X’ and ‘Y’ are variables and ‘m’ is Slope and ‘c’ is a constant.

2. **Point Slope form** of the linear equation can be represented as:

Y - Y$_1$ = m(X - X$_1$);

In above equation (X$_1$, Y$_1$) are two points on line.

3. **Two point form** of the equation is:

Y - Y$_1$ = [(Y$_2$ - Y$_1$) / (X$_2$ - X$_1$)] (X - X$_1$);

Here (X$_1$, Y$_1$) and (X$_2$, Y$_2$) are two points on line.4. **Intercept form** can be represented as:

X/a + Y/b = 1;

Here ‘a’ and ‘b’ are two constants and are denominator of ‘X’ and ‘Y’ and may have different values.

To solve linear equations of two variables the method of substitution and method of elimination are the basic methods.

Let us discuss some of the word problems. While solving word problems we should proceed the following steps.

Step 1: Read the problem carefully and and assume the the unknowns say x and y respectively.

Step 2: Frame the equations according to the given condition and write it in the the general form ax + by + c = 0.

Step 3: Solve the two equations for the variables x and y by Substitution method or Elimination Method (preferred)

Step 4: Write the final answer of the question as given in the problem.

Example 1: A school has organized an educational trip to national Bal Bhavan, Delhi. 148 students went for the trip. There were ten drivers and two types of vehicles including cabs and buses. Each cab has 10 seating capacity and each bus has 22 seating capacity, including driver seat. How many buses and cabs did school hired for the educational trip?

Solution: Let company hired x number of buses and y number of cabs.

Total number of drivers = 10

x + y = 10 .....(1)

Number of employees going for city trip = 148

22x + 10y = 148 ....(2)

Find the value of y from (1) and substitute in (2)

(1)=> y = 10 - x ....(3)

Now 22x + 10(10 - x) = 148 (Substitution method)

22x + 100 - 10x = 148

12x = 48

x = 4 (on solving)

Put x = 4 in (3), we get

y = 10 - 4 = 6

Company hired 4 buses and 6 cabs.

Example 2: Shekhar's father is 3 times as old as Shekhar. Hence 7 years, Shekhar's father will be two times as old as Shekhar. Find their present ages.

Solution:

Since after 7 years, Shekhar's father will be two times as old as Shekhar, this implies

3x + 7 = 2(x + 7)

3x + 7 = 2x + 14

x = 7

So 3x = 3 $\times$ 7 = 21

Therefore, Shekhar is 7 years old and his father is 21 years old.

Example 2: Shekhar's father is 3 times as old as Shekhar. Hence 7 years, Shekhar's father will be two times as old as Shekhar. Find their present ages.

Solution:

Shekhar | Shekhar's father | |

Present Age | x | 3x |

After 7 Years | x + 7 | 3x + 7 |

Since after 7 years, Shekhar's father will be two times as old as Shekhar, this implies

3x + 7 = 2(x + 7)

3x + 7 = 2x + 14

x = 7

So 3x = 3 $\times$ 7 = 21

Therefore, Shekhar is 7 years old and his father is 21 years old.

First condition: the sum of 2 numbers is 20 and

Second condition: one of the numbers exceeds the other by 7.

**Question 3**: Rekha added 150 to a number and recorded result as 74 which is more than 2 times the number. Find the number.