How to create and solve Linear and quadratic equations
Before you study the NCERT textbook of tenth class maths, the linear equations, and quadratic equations, you have to understand how to create and solve algebraic equations from a given algebraic question. The linear equations and quadratic equations both are in the curriculum of X class of every board weather it is CBSE, ICSE or any other state board of the world. In CBSE board almost 25 marks of the total 80 marks questions are asked in the final board exam of X class from the chapters of linear equations in two variables and quadratic equations. Future Study Point is introducing here few tips of solving quadratic and pair of linear equations through the solution of a few examples, once you go through the whole of the post you might become an expert in both of the lessons and you would be capable to grab hundred percent marks in the questions asked from the lessons of linear equations and quadratic equations.
After you go through this post on linear and quadratic equations you will become confident and you could solve all the unsolved questions of linear and quadratic equations of your textbook like NCERT or of other guides like R.D Sharma. Here each question of linear and quadratic equations is literally explained step by step in such a way that you could clear all your doubts in building a pair of linear equations or quadratic equations.
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How to create a pair of linear equations
Pair of linear equation in two variables→ In these questions you are given two statements in the questions in which two variables are related to each other. From these statements, you have to build up equation 1 and equation 2, thereafter you can evaluate both of the variables presented in the pair of linear equations. Here the solution of the linear equations is shown algebraically and not by the way of the graph because our main focus is to let you understand how to build a pair of the equations. Study the following examples.
Q1. Sita tells his daughter that 7 years ago she was 7 times as old as she were and after 3 years then she should be one-third of her age. Find the present ages of both.
Ans. In this question the first statement is 7 years ago Sita was 7 times as old as her daughter.
Here the age of Sita is given in terms of her daughter, so it is convenient to write the age of Sita(x) in the left part of the equation and in the right part write the age of her daughter(y)
7 years back, their ages
7 years back Sita’s age = 7 × 7 years back her daughter’s age
x – 7 = 7y – 49
x – 7y = –42……….(1)
The second statement of the question is after 3 years Sita’s daughter becomes one-third of Sita’s age or Sita becomes thrice of her daughter’s age (because it is simple to write).
Sita’s age after 3 years= 3 × Her daughter’s age after 3 years
x + 3 = 3y + 9
x – 3y = 6………….(2)
x – 7y = –42……….(1)
Subtracting one eq.(1) from eq.(2)
We get y = 12, putting the value of y in equation (2), we get x = 42
Hence the age of Sita is 42 years and the age of her daughter is 12 years
Ex2. 5 men and 2 women do a piece of work in 4 days and 6 men and 3 women can do that work in 3 days . How many days a man or a woman alone can do the same work.
Ans. First of all, suppose the variables, you are asked in the question, here we have to suppose two-variable the number of days a man alone take to complete the work (x) and number of days a woman alone take to complete the work(y).
First statement of the question is 5 men and 2 women do a piece of work in 4 days
5× 1 day work of a man + 2 × 1 day work of a woman = 1 day work of 5 men and 2 women
Let’s find the one day work of man and of woman and 1 day work of 5 men and 2 women as follows.
In x day a man alone will do → 1 piece of work
In 1 day a man will do →
In y days a woman alone do → 1 piece of workI
In 1 day a woman alone will do→ 1/y
5 men and 2 women do in 4 days → 1 piece of work
5 men and 2 women do in 1 day → 1/4
Therefore the linear equation according to first statement is as follows
5/x + 2/y = 1/4……..(1)
Following the same procedure in building equation (2) from the second statement of the question.
6/x + 3/y = 1/3……..(2)
Multiplying (1) by 6 and eq.(2) by 5 ,we get the following equations (3) and (4)
30/x + 12/y = 3/2 ……(3)
30/x + 15/y = 5/3
Subtracting (4) from (3) we have
–3/y = –1/6
y = 18, putting this value in equation (1)
5/x + 2/18 = 1/4
5/x = 1/4 – 2/18
5/x = (9 –4)/36
x = 36
Therefore a woman alone can do that piece of work in 18 days and a man alone can do that work in 36 days.
How to create and solve the quadratic equation
QUADRATIC EQUATION- The equation in the form of ax² + bx + c =0 is known as a quadratic equation, there are certain questions which can be solved by the formation of either the system of linear or quadratic equations, few of the examples are given below.
Quadratic equations can be solved by the factorization method, complete squire method or by the quadratic formula.
Read our post the solution of quadratic equation by the complete squire method
⇒Generally when product of two variable is given or manipulation of both statements become possible to form a single quadratic equation, then it is convenient for the students to solve it by the rule of the quadratic equation. See the examples.
Ex3. The product of two consecutive natural number is 420,find the numbers.
Ans. Let first number is x and second will be x +1
According to question x(x +1) = 420
x² + x –420 = 0
x² + 21x –20x –420 =0
x(x+ 21) –20(x + 21) = 0
(x+ 21)(x –20) = 0
x = 20,–21
Avoiding the number in negative sign
Therefore the natural numbers are x =20, x +1=21
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