**How to create and solve Linear and quadratic equations**

**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**.

**NCERT Solutions Class 10 Science from chapter 1 to 16**

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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**NCERT Solutions Class 10 Science from chapter 1 to 16**

**NCERT Solutions of all chapters of Maths for Class 10 from Chapters 1 to 15**

**NCERT Solutions of Class 9 Science : Chapter 1 to Chapter 15**

**NCERT Solutions of Class 9 Maths : from chapter 1 to 15**

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**Class 10 maths NCERT solutions of chapter 7- Coordinate Geometry**

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**Solutions of Class 10 maths question paper Set-3 CBSE Board 2019**

**CBSE Class 10 science question paper 2020 SET -3 solutions**

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**How to create and solve Linear and quadratic equations**

**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.

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**How to create and solve Linear equations**

**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)

x → y

7 years back, their ages

x – 7 and y – 7

7 years back Sita’s age = 7 × 7 years back her daughter’s age

x – 7 = 7( y- 7)

Simplifying it

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).

x → y

After 3 years their ages are

x + 3 and y + 3

Sita’s age after 3 years= 3 × Her daughter’s age after 3 years

x + 3 = 3(y + 3)

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 → (1/x) part of the work

In y days a woman alone do → 1 piece of work

In 1 day a woman alone will do→ (1/y) part of the work

According to first condition,5 men and 2 women do in 4 days → 1 piece of work

5 men and 2 women will do in 1 day →(1/4) Part of the work

Therefore the linear equation according to first statement is as follows

1 day work of 5 men = 5 × 1 day work of a man = 5× 1/x = 5/x

1 day work of 2 women = 2× 1 day work of a woman = 2× 1/y = 2/y

Therefore,we have

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 Equations.** 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

**Finding the roots of the polynomial by the complete square 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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**NCERT Solutions of Science and Maths for Class 9,10,11 and 12**

**NCERT Solutions of class 9 maths**

Chapter 1- Number System | Chapter 9-Areas of parallelogram and triangles |

Chapter 2-Polynomial | Chapter 10-Circles |

Chapter 3- Coordinate Geometry | Chapter 11-Construction |

Chapter 4- Linear equations in two variables | Chapter 12-Heron’s Formula |

Chapter 5- Introduction to Euclid’s Geometry | Chapter 13-Surface Areas and Volumes |

Chapter 6-Lines and Angles | Chapter 14-Statistics |

Chapter 7-Triangles | Chapter 15-Probability |

Chapter 8- Quadrilateral |

**NCERT Solutions of class 9 science **

**CBSE Class 9-Question paper of science 2020 with solutions**

**CBSE Class 9-Sample paper of science**

**CBSE Class 9-Unsolved question paper of science 2019**

**NCERT Solutions of class 10 maths**

**CBSE Class 10-Question paper of maths 2021 with solutions**

**CBSE Class 10-Half yearly question paper of maths 2020 with solutions**

**CBSE Class 10 -Question paper of maths 2020 with solutions**

**CBSE Class 10-Question paper of maths 2019 with solutions**

**NCERT solutions of class 10 science**

**Solutions of class 10 last years Science question papers**

**CBSE Class 10 – Question paper of science 2020 with solutions**

**CBSE class 10 -Latest sample paper of science**

**NCERT solutions of class 11 maths**

Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |

Chapter 7- Permutations and Combinations | Chapter 15- Statistics |

Chapter 8- Binomial Theorem | Chapter 16- Probability |

**CBSE Class 11-Question paper of maths 2015**

**CBSE Class 11 – Second unit test of maths 2021 with solutions**

**NCERT solutions of class 12 maths**

Chapter 1-Relations and Functions | Chapter 9-Differential Equations |

Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |

Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |

Chapter 4-Determinants | Chapter 12-Linear Programming |

Chapter 5- Continuity and Differentiability | Chapter 13-Probability |

Chapter 6- Application of Derivation | CBSE Class 12- Question paper of maths 2021 with solutions |

Chapter 7- Integrals | |

Chapter 8-Application of Integrals |

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