SUBJECT- MATHEMATICS CLASS-VIII TOPIC- LINEAR EQUATIONS IN ONE VARIABLE

In this chapter, students explore algebraic expressions, learn about linear equations in one variable, solve using cross multiplication method, and apply them in real-life scenarios. Examples include buying chocolates and car rental charges. The content also covers forming equations, terms in expressions, and the concept of equality. Included are learning objectives, PDF chapter link, and a video on linear equation applications.

• Linear Equations
• Algebraic Expressions
• Cross Multiplication
• Real-Life Applications
• Learning Objectives

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1. SUBJECT- MATHEMATICS CLASS-VIII TOPIC- LINEAR EQUATIONS IN ONE VARIABLE Work is Worship

2. PDF CHAPTER LINK(LINEAR EQUATIONS IN ONE VARIABLE) https://s.docworkspace.com/d/APktL_TYhNxFgcXzj8edFA Work is Worship

3. LEARNING OBJECTIVES In this chapter students will be able to know What is an algebraic expression What is linear equation Methods of solution of linear equations in one variable (cross multiplication method) Application of linear equation in one variable in our daily life. Solving word problems by framing linear equations Work is Worship

4. Linear equation story (APPLICATION ) WATCH THE VIDEO PLAY PLAY Work is Worship

5. REAL LIFE EXAMPLES OF APPLICATIONS LINEAR EQUATIONS Some examples where linear equations are used in our daily life are 1.Suppose for your birthday you want to buy chocolates from the market .The cost of one chocolate is Rs.20. You have to buy n chocolates .Then You have to pay Rs.20n. So it can be written in equation form a=20n ,where a is amount of money you have to spend to buy n chocolates . Work is Worship

6. REAL LIFE EXAMPLES OF APPLICATIONS LINEAR EQUATIONS 2. you take a car for rent .They have a fixed charge of Rs.100 Plus Rs.40 for every hour. This can be framed in an equation as Y=40x+100 where x is hours used and y is the money you have to pay . Work is Worship

7. ALGEBRAIC EXPRESSIONS An algebraic expression is a combination of constants and variables connected by means of four fundamental operations addition, subtraction, multiplication and division(+,- ,x ,/). For example 3x+5. 6a3 b+9x,10y3 , xyz/p are algebraic expressions. Parts of an algebraic expressions separated by symbols are called terms of an algebraic expression. An equation is formed when two algebraic expressions are joined together with an equality sign. So 6x+5y= 9xy is an equation, where two algebraic expressions 6x+5y and 9xy are Equation- A statement of equality which contains one or more unknown quantity or variable is called an equation. 3x+2=5, 6x+xy=4 are equations. joined with equality sign. Work is Worship

8. INTRODUCTION TO LINEAR EQUATIONS IN ONE VARIABLE Watch the video: PLAY PLAY Work is Worship

9. INTRODUCTION TO LINEAR EQUATION IN ONE VARIABLE Linear equations in one variable-An equation that contains only one variable and the highest power of the variable is one equation in For example : 4x-6=15, (3x/7)=5, (x+3)/(7x-3)=3 are linear equations in variable x. is called a linear one variable. Work is Worship

10. INTRODUCTION TO LINEAR EQUATIONS IN ONE VARIABLE The equation of the form ax+b=c, where a , b and c are rational numbers and a 0, is called a linear equations in one variable. The value of the variable which satisfies the given expression is called a solution or root of the equation. The standard form of the linear equation in one variable is px+q=0, where p and q are rational numbers and p 0.Its solution is given by x= - (q/p) Work is Worship

11. RULES FOR SOLVING LINEAR EQUATIONS IN ONE VARIABLE RULE-1- Same quantity can be added to both side Rules For Solving Linear Equations Rules s of an equation without changing the quality. Rule-2- same quantity can be subtracted from both sides of an equation without changing the equality. Rule-3- Both sides of an equation may be multiplied by the same non-zero number without changing the equality Rule-4- Both sides of an equation may be divided by the same non-zero number without changing the equality. Some complicated equations can be solved by using two or more of these rules together. Work is Worship

12. Activity on linear equation in one variable Watch the video PLAY PLAY PLAY Work is Worship

13. SOLVING EQUATIONS OF THE FORM (ax+b)/(cx+d)=k Example-1-Solve (2x-1)/(3x+5) =5 and verify your answer. (3) Answer- The given equation is not a linear equation. so we will convert it to linear equation. We multiply both sides by (3x+5) (2x-1)/(3x+5)X (3x+5)= 5 X (3X+5) (0.5) or,2x-1=5(3x+5) or,2x-1=15x+25 (0.5) or, 2x-15x=25+1 (0.5) or, -13x=26 or, x=26/-(13)=-2 (0.5) Verification- Putting x=-2 in L H S of the given equation LHS={(2X(-2)-1}/{3X(-2)+5}=-5/-1 =5=RHS hence x=-2 is the solution. (1) Work is Worship

14. CROSS MULTIPLICATION METHOD In this method , the given steps are followed if the equation is of the form (ax+b)/(cx+d) =k , where cx+d 0 1.Multiply the numerator of LHS by the denominator of RHS. 2.Multiply the numerator of RHS by the denominator of LHS. 3. Equate the expressions obtained in (i) and (ii). Work is Worship

15. SOLVING EQUATIONS BY CROSS MULTIPLICATION METHOD EXAMPLE-2 Solve the equation (7y-5)/(4y+2)=8/7 and verify your answer.(3) Answer : Cross multiplying we get, 7(7y-5)=8(4y+2) (0.5) or, 49y-35=32y+16 (0.5) or, 49y-32=16+35 (0.5) or, 17y=51 so y=51/17 =3 (0.5) Verification-For y=3 L H S= (7y-5)/(5x+1)=(7X3-5)/(12+2)=16/14=8/7=RHS so , y=3 is the solution. (1) Work is Worship

16. CROSS MULTIPLICATION METHOD Example-3-Find the positive value of x which satisfies the equation , (x2 + 5)/(2-x2 ) = -3/2 (3) solution: Put x2 =y in the given equation. Then , it becomes (y+5)/(2-y)=-3/2 (0.5) By cross multiplying , we get, 2(y+5)=-3(2-y) or, 2y+10= -6 +3y (0.5) or, 2y-3y=-6-10 or, -y=-16 (0.5) so y=16 Since y=x2 so, x2 =16=42 or x=4 (0.5) Verification-For x=4 L H S= (x2 +5)/(2-x2 ) =(42 +5)/(2-42 )=(16+5)/(2-16) =21/-14=-3/2=RHS Hence, x=4 is the solution. (1) Work is Worship

17. ANSWER THE FOLLOWING QUESTIONS 1.Solve the following equations and verify your answer. (i) (2x-1)/5x = -1/6 (ii)(3k+5)/(4k-3)=4/9 (iii) (4z-3)/(2z+1)=5/7 2.Find the positive value of the variable for which the given equation is satisfied. (i) (3-x2 )/(8+x 2 )=-3/4 (ii)(x2 9)/(5+x2 )=5/9A Work is Worship

18. APPLICATIONS OF LINEAR EQUATIONS The following steps should be followed to solve a word problem. Step-1-Read the problem carefully and note what is given and what is required. Step-2-Denote the unknown quantity by some letters, Say x,y,z etc Step-3-Translate the statement of the problem into mathematical statements step-4-Using the conditions given in the problem,find the equation. Step-5-Solve the equation for the unknown. Step-6-check whether the solution satisfies the equation Work is Worship

19. WORD PROBLEMS ON LINEAR EQUATION Example-4- The present ages of Ram and Rahim are in the ratio 4:3. Four years later , Their ages will be in the ratio 6:5.What are there present ages.(3) Answer-Let present age of Ram be 4x and years and Rahim s be 3x years. After 4 years, Ram s age =(4x+4)years Rahim s age=(3x+4)years (1) According to given condition, (4x+4)(3x+4) =6:5 or, 5(4x+4)=6(3x+4) (0.5) or, 20x+20=18x+24 or, 20x-18x=24-20 or, 2x=4 or, x=2 (1) so Present age of Ram = (4X2)years=8years Present age of Rahim=(3X2)years=6years (0.5) Work is Worship

20. WORD PROBLEMS ON LINEAR EQUATION Example-5-The sum of the digits of a two digit number is 12. The number obtained by interchanging the digits exceeds the original number by 54. Find the original number. (3) Answer-Let the digit at ones place be x so the digit at tens place=12-x Thus the original number=10X(12-x)+x =120-10x+x=120-9x (0.5) on interchanging the digits of the given number, the digit at ones place becomes (12-x) and digit at tens place becomes x. New number=10x+(12-x)=9x+12 (0.5) according to question New number original number=54 (9x+12)-(120-9x)=54 9x+12-120+9x=54 or, 18x=54+108 or, 18x=162 or, x=162/18=9 (1) digit at ones place=9 The digit at tens place = 12-9=3 so original number=39 (1) Work is Worship

21. WORD PROBLEMS ON LINEAR EQUATION Example-6- The sum of three consecutive multiples of 8 is 888. find these multiples.(3) Answer-Let the first multiple of 8 be 8x. Then, the next two multiples of 8 will be 8(x+1) and 8(x+2). (0.5) It is given that the sum of these three consecutive multiples is 888. so 8x+ 8(x+1)+8(x+2)=888 (0.5) or, 8x+8x+8 +8x+16=888 (0.5) or, 24x+24=888 or, 24x=888-24 or, 24x=864 (0.5) or, x=864/24=36 (0.5) so the three consecutive multiples are 8X36,8X37 and 8X38 so the numbers are 288,296 and 303. (0.5) Work is Worship

22. WORD PROBLEMS ON LINEAR EQUATION Example-7-A steamer goes down stream from one port to another in 6 hours. It covers the same distance upstream in 7 hours.If the Speed of the stream be 2km/hr, find the speed of the steamer in still water.(4) ANSWER-Let the speed of the steamer in still water be xkm/hr Given speed of the stream=2km/hr speed of steamer in downstream=(x+2)km/hr Distance covered in 1 hour=(x+2)km (1) Distance covered in 6 hour=6(x+2)km So distance between two ports=6(x+2)km Similarly speed of the steamer in upstream= (x-2)km/hr So distance covered in 7 hours=7(x-2)km (1) hence 6(x+2)=7(x-2) (0.5) or, 6x+12=7x-14 (0.5) or, 6x-7x=-14-12 or, -x=-26 or ,x=26 (1) so speed of steamer in still water is 26km/hr Work is Worship

23. WORD PROBLEMS ON LINEAR EQUATION Example-8 -The perimeter of a rectangle is 100m.If the length is decreased by 2m and the breadth is increased by 3m,the area increases by 44 m2 .Find the length and breadth of the rectangle. (4) Answer-Let length of the rectangle be x m . According to question 100=2(x+ breadth) so x+ breadth=50 (O.5) breadth=(50-x)m Area of new rectangle=(x-2)(50-x+3)=(x-2)(53-x) (1) according to question, (x-2)(53-x)-x(50-x)=44 (0.5) or, 53x-x2 -106+2x-50x+x2 =44 (0.5) or 5x=44+106 or, 5x=150 or, x=150/5=30 (1) so length of the given rectangle=30m breadth of the rectangle=(50-30)m=20m (0.5) so length and breadth of the rectangle are 30m and 20m respectively. Work is Worship

24. ANSWER THE FOLLOWING QUESTIONS 1. Two years ago, father was three times as old as his son and two years hence, twice his age will be equal to five times that of his son. Find their present ages. 2.The sum of the digits of a 2-digit number is 8. The number obtained by interchanging the digits exceeds the given number by 18. Find the given numbers. 3.A motor boat goes downstream and covers a distance in 4 hours while it covers the same distance upstream in 5 hours. If the speed of the stream is 3km/hr, find the speed of the motor boat in still water. Work is Worship

25. LEARNING OUTCOMES The Students are able to know : 1. What is a linear equation in one variable 2. Rules for solving a linear equation. 3. The method to solve equations of the form (ax+b)/(cx+d)=k. 4. Process to formulate the word problems and solve these. Work is Worship

26. CONCEPT MAP METHODS OF SOLUTION APPLICATIONS LINEAR EQUATION IN ONE VARIABLE CROSS MULTIPLICATION METHOD WORD PROBLEMS GENERAL FORM ax+b=0, a 0 SOLVE (2X+6)/(3X-1) =2 FRAMING OF EQUATIONS 2x+6=2(3x+1) SOLUTION X=-b/a 2x+6=6x+2 SOLVING EQUATION -4x=-4 2x-6x=2-6 x=1 Work is Worship

27. EQUATIONS QUOTE If A is a success in life, then A equals x plus y plus z. Work is x; y is play ; and z is keeping your mouth shut A=x+y+z Albert Einstein Work is Worship

28. THANK YOU Work is Worship