Application Of Linear Equations In One Variable
A statement which states that the two algebraic expressions are equal
, it is called an equation. e.g. Each of the following algebraic statement is an equation.
(i) 9x2 +8 = x 3, (ii) 3x + 4y = 8, (iii) x 3 = 3x + 8, etc.
The equation involving only one variable (unknown) in first order
(that is with highest power equal to one) is called a linear equation.
eg. 3x 5 = 0, 8 y = 15, 7 + 3z = 10, etc.
An equation remains unaltered (unchanged) on:
(i) adding the same number to both sides of it.
(ii) subtracting the same number from both sides of it.
(iii) multiplying both sides of it by the same number; and
(iv) dividing both sides of it by the same number.
To solve Problems based on Linear Equations we need to keep in mind the below steps:
Steps: 1. Read the problem carefully to know what is given and what is to be found.
2. Represent the unknown quantity as x or some other letter such as a, b, y, z, etc.
3. According to the conditions, given in the problem, write the relation between
the given quantity and quantity to be found.
4. Solve the equation to obtain the value of the unknown.
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Example Problems on the Application of Linear Equation in One Variable:
Ex 1: The difference of the squares of two consecutive even natural numbers is 92. Taking x as the smaller of the two numbers, form an equation in x and hence find the larger of the two.
Solution: Since, the consecutive even natural numbers differ by 2 and it is given that the smaller of the two numbers is x; therefore, the next larger even number is x + 2.
According to the given statement: (x + 2)2 x2 = 92 [Difference of the squares]
implies x2 + 4x + 4 x2 = 92
implies 4x = 92 4 = 88.
implies x = 22.
Therefore Larger even number = x + 2 = 22 + 2 = 24.
Ex 2: A man is 24 years older than his son. In 2 years, his age will be twice the age of his son. Find their present ages.
Solution: Let the present age of the son by x years.
Therefore Present age of the father = (x + 24) years.
In 2 years: The mans age will be (x + 24) + 2 = (x + 26) years
and sons age will be x + 2 years
According to the problem: x + 26 = 2 (x + 2)
On solving we get: x = 22.
Therefore Present age of the man = x + 24 = 22 + 24 = 46 years
and, Present age of the son = x = 22 years.
Ex 3: Two consecutive even numbers are such that half of the larger exceeds one-fourth of the smaller by 5. Find the numbers.
Solution: Let the required even numbers be x and x + 2.
Given: 1/ 2 (x + 2) (1/4) x = 5.
= 2x + 4 - x] / 4 = 5.
= x + 4 = 20 that is x = 20 4 = 16.
Therefore, the required numbers are = x and x + 2.
= 16 and 16 + 2 .
implies The numbers are 16 and 18.
Practice Problems on the Application of Linear Equations in One Variable:
1. The length of a rectangle is twice its width. If its perimeter is 54 cm, find its length.
Rectangle Problem
[Ans: 10 cm]
2. The difference of two numbers is 3 and the difference of their square is 69. Find the numbers.
[Ans: 13 and 10]
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