Today our topic of discussion is Usage of linear equations.
Usage of linear equations
Usage of linear equations
In real life we have to solve different types of problems. In most cases of solving these problems mathematical knowledge, skill and logic are necessary. In real cases, in the application of mathematical knowledge and skill, as on one side the problems are solved smoothly, on the other side in daily life, solutions of the problems are obtained by mathematics.
As a result, the students are interested in mathematics. Here different types of problems based on real life will be expressed by equations and they will be solved.
For determining the unknown quantity in solving the problems based on real life, variable is assumed instead of the unknown quantity and then equation is formed by the given conditions. Then by solving the equation, value of the variable, that is the unknown quantity is found.
Example 5. The digit of the units place of a number consisting of two digits is 2 more than the digit of its tens place. If the places of the digits are interchanged, the number thus formed will be less by 6 than twice the given number. Find the number.
Solution: Let the digit of tens place be z.Then the digit of units place will be x + 2
the number is 10x + (x + 2) *
or, 11x + 2
Now, if the places of the digits are interchanged, the changed number will be
10(x + 2) + xor, 11x + 20
By the question, 11x + 20 = 2(11x + 2) – 6
or, 11x + 20 = 22x + 4 – 6
or, 22x – 11x = 20 + 6 – 4 by transposition]
or, 11x = 22
alpha*r, x = 2
∴ the number is 11x + 2 = 11 * 2 + 2 = 24
∴given number is 24
Example 6. In a class if 4 students are seated in each bench, 3 benches remain vacant. But if 3 students are seated on each bench, 6 students are to remain standing. What is the number of students in that class?
Solution: Let the number of students in the class be
Since, if 4 students are seated in a bench, 3 benches remain vacant, the number of benches of that class = x/4 + 3
Again, since, if 3 students are seated in each bench, 6 students are to remain standing, the number of benches of that class- 3
Since the number of benches is fixed,
x/4 + 3
= (x – 6)/3 (x + 12)/4
= (x – 6)/3 or, 4x – 24
= 3x + 36or , 4x – 3x
= 36 + 24
therefore,
or, x = 60
∴number of students of the class is 60
Example 7. Mr. Kabir, from his Tk. 56000, invested some money at the rate of profit 12% per annum and the rest of the money at the rate of profit 10% per annum. After one year he got the total profit of Tk. 6400. How much money did he invest at the rate of profit 12%?
Solution: Let Mr. Kabir invest Tk. z at the rate of profit 12%. ..he invested Tk. (56000-) at the rate of profit 10%
Now, profit of Tk. a in 1 year is Tk. x * 12/100 or Tk.
(12x)/100
Again, profit of Tk. (56000) in 1 year is Tk. or, Tk, (10(56000 – x))/100 (56000 – x) * 10/100
By the question, (12x)/100 + (10(56000 – x))/100 = 6400
or, 12x + 560000 – 10x = 640000
or, * 0.2x = 640000 – 56000
or, * 0.2x = 80000
or, x = 40000
Mr. Kabir invested Tk. 40000 at the rate of profit 12%.
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