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# LINEAR EQUATIONS

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1. LINEAR EQUATIONS Mrs. Chanderkanta -9x - 4x = -36 9x - 4x = -36 3x-4y =7 3x –7y =21 Mrs. Anju Mehta 5x –8y =-40 -6x +7y = 42 2x+ 3y =6 3x –7y =21

2. Target group Class ninth and tenth

3. LEARNING OBJECTIVES • Define the linear equation in two variable. • Solution of linear equation. • Converts a linear equation of two variable in graphical form . • Solve simultaneous linear equation by graphical method. • Learn computer skills. • Learn about MS Office. • Develop a habit of research. • Learn to insert the pictures and relevant text in their presentation . • Learn editing skill.

4. WHEN we talk to each other, we use sentences. What do we say? Either we talk or we give some statements These statements may be RIGHT or WRONG For example we make the statement- ”sunrises in the east and sets in the west”

5. WHEN we talk to each other, we use sentences. What do we say? Either we talk or we give some statements These statements may be RIGHT or WRONG For example we make the statement- ”sunrises in east and sets in west” This is a TRUE statement It is not necessary that all the statements are true. Some are true and some are false. In mathematics we call those statements as OPEN STATEMENTS

6. If an open statement becomes TRUE for some value then it is called EQUALITY and it is represented by the sign “=“ An EQUALITY has two sides L.H.S. and R.H.S. where, L.H.S. = R.H.S.

7. In mathematics, we often use OPEN STATEMENTS For example the statement , “ any number added to 5 will give 8” is an open statement If we add any number to 5, we may or may not get 8 5 + 1= 8 FALSE STATEMENT FALSE STATEMENT 5 + 2 = 8 5 + 3 = 8 TRUE STATEMENT The number 3 makes both the sides equal. Hence the statement becomes TRUE.

8. 2 kg 5 kg How much weight should be added to equalize the balance? + 2 kg = 5 kg

9. + 2kg = 5kg The above statement becomes x+ 2= 5 This statement is called an EQUATION This equation will be true depending on the value of the variable ‘x’

10. So we can say, ax+b = 0 is an equation in one variable x Where a,b are constants & a = 0

11. Let us take an example from daily life. Cost of two rubbers and three pencils is six rupees In mathematical form, it can be written as 2x + 3y = 6, where x is the cost of one rubber and y of one pencil (3, 0) (0,2) Ordered pairs

12. Let us plot the ordered pairs: (3,0) Show me (0,2) Show me Y- axis 3 (0,2) 2 * 2x + 3y =6 1 (3,0) * 0 -3 -2 -1 1 2 3 4 5 6 7 -1 x-axis -2 -3

13. You have seen that the equation 2x+3y =6 is giving a straight line in the graph Note: Solutions of an equation 2x + 3y =6 are x =0 , y=2 and x=3 , y=0. In any equation of the type ax + by+ c = 0 where a, b, c --- constants x , y --- variables will gives straight line in the graph These types of the equations are called LINEAR EQUATIONS

14. If in an equation ax+ by + c= 0 Case1: When a =0,b= 0, then 0x + by +c = 0 e.g. in an equation 2x+3y =6 , If a=0 0x + 3y =6 3y = 6 –0x y =6-0x 3

15. Let us plot the ordered pairs: (-3,2) (1,2) (2,2) Show me Show me Show me Y- axis LINE IS PARALLEL TO X-AXIS 3 0x+3y =6 * 2 * * (-3,2) (1,2) (2,2) 1 0 -3 -2 -1 1 2 3 4 5 6 7 -1 x-axis -2 -3

16. Case2: if in an equation ax+ by + c= 0 when a =0, b =0, then ax + 0y + c =0 e.g. in an equation 2x+0y =6 , when b=0 2x + 0y =6 2x = 6 – 0y x =6-0y 2

17. Let us plot the ordered pairs: (3,3) (3,-2) (3,1) Show me Show me Show me Y- axis LINE IS PARALLEL TO Y-AXIS 2x +0y =6 3 * (3,3) 2 (3,1) * 1 0 -3 -2 -1 1 2 3 4 5 6 7 -1 x-axis -2 (3,-2) * -3

18. if in an equation ax+ by + c= 0 when Case3: When a =0,b= 0, c =0 ax +by = 0 e.g. in an equation 2x+3y =6 , if c=0 2x + 3y =0 2x = -3y x =-3y 2

19. Let us plot the ordered pairs: (0,0) (-3,2) Show me Show me (3,-2) Show me Show me Y- axis LINE PASSES THROUGH THE CENTER 3 (-3,2) * 2 1 (0,0) * 0 -3 -2 -1 1 2 3 4 5 6 7 -1 x-axis (3,-2) -2 * -3 2x+3y =0

20. If we draw two linear equations in one graph then we have three possibilities: one solution 1: Intersecting lines * 2: Parallel lines no solution 3. Lines will coincide many solutions

21. Now there is an exercise for you. Take any two linear equations. Plot them on the graph and observe what type of solution you get.

22. ACKOWLEDGEMENT • Mr. V.K. Sodhi ,Senior Lecturer,S.C.E.R.T. • “Mathematics” by R.S.AGGARWAL • N.C.E.R.T. BOOK FOR Mathematics for Class-X Internet sites: www.math.nice.edu www.math.org.uk www.pass.math.org.uk