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Maths

1. 1. Linear Equations in Two Variables MADE BY : 1. Shekhar 2. Suhail khan 3. Hansika 4. Nihal 5. Akash kr. 6. Babita 7. Kanchan IXth –C
2. 2. Equations of the form ax + by = c are called linear equations in two variables.Where A, B, and C are real numbers and A and B are not bothzero. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept. x y 2-2 This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) The graph of any linear equation in two variables is a straight line
3. 3. Equations of the form 𝒂𝟏 𝒃𝟐 ≠ 𝒃𝟏 𝒃𝟐 are called INTERSECTING LINES These type of equations have Exactly one solution(unique) These are consistent lines Example: x-2y=0 3x+4y=20 𝑎1 𝑎2 = 1 3 𝑏1 𝑏2 = −2 4 𝑎1 𝑎2 ≠ 𝑏1 𝑏2
4. 4. Equations of the form 𝒂𝟏 𝒃𝟐 = 𝒃𝟏 𝒃𝟐 = 𝒄𝟏 𝒄𝟐 are called COINCIDENT LINES These type of equations have Infinity many no. of solution. These are consistent lines Example: 2x+3y=9 4x+6y=18 𝑎1 𝑎2 = 1 2 𝑏1 𝑏2 = 1 2 𝑎1 𝑎2 = 𝑏1 𝑏2 = 𝑐1 𝑐2 𝑐1 𝑐2 = 1 2
5. 5. Equations of the form 𝒂𝟏 𝒃𝟐 = 𝒃𝟏 𝒃𝟐 ≠ 𝒄𝟏 𝒄𝟐 are called PARALLEL LINES These type of equations have no solution. These are inconsistent lines Example: x+2y=4 2x+4y=12 𝑎1 𝑎2 = 𝑏1 𝑏2 ≠ 𝑐1 𝑐2 𝑎1 𝑎2 = 1 2 𝑏1 𝑏2 = 1 2 𝑐1 𝑐2 = 1 3
6. 6. EXAMPLE : X + y=10 X - y=4 x 5 6 7 y 5 4 3 x 3 2 5 y -1 -2 1
7. 7. Example: x + y=5 2x+2y=10 x 0 2 1 y 5 3 4
8. 8. Example: 2x + y=160 2x+y=150 x 60 70 50 y 40 20 60 x 60 50 40 y 30 50 70
9. 9. EXAMPLE : x + y=14 x – y=14 Solution : x + y=14-------------------------1 x – y=14-------------------------2 From equation 1 x =14-y Put the value of x in equation 2 14-y-y=4 14-2y=4 14-4=2y 10=2y Y=5 x =14-5 X=9
10. 10. EXAMPLE : x + y=25 x + 2y=40 x + y=25-------------------------1 x + 2y=40-------------------------2 Solution : Subtract equation 2 from 1 x + 2y=40 - - - x + y=25 -y=-15 -y=-15 Y=15 x + 15=25 X=25-15 X=10
11. 11. The general form of cross- multiplication method is: 𝒙 𝒃𝟏𝒄𝟐−𝒃𝟐𝒄𝟏 = 𝒚 𝒄𝟏𝒂𝟐−𝒄𝟐𝒂𝟏 = 𝟏 𝒂𝟏𝒃𝟐−𝒂𝟐𝒃𝟏
12. 12. LINEAR EQUATION SYSTEM with 2 variables
13. 13. DO YOU REMEMBER?  Which one of the following example is linear equation of one variable? Give your reason. 2 1 a) (4 2) 3 2 b) 3 2 5 0 c) 3 2 7 2 d) 2 3 x x x y y x y           The point (a) and (c) are examples of linear equation of one variable. Can you find the solution?
14. 14.  If I have what the meaning of solution of that system?  In how many ways we can solve linear equation system? we can solve linear equation system in four ways, that are substitution, elimination, substitution-elimination and graph method 2 4 2 3 12 x y x y    
15. 15. LINEAR EQUATION SYTEM WITH 2 VARIABLES  In general, a linear equation system of 2 variables x and y can be expressed:  Let’s try to solve this problem 1 1 1 2 2 2 a x b y c a x b y c     1 1 1 2 2 2where , , , , , anda b c a b c R 2 4 2 3 12 x y x y      ( , )SS x y
16. 16. LINEAR EQUATION SYTEM WITH 2 VARIABLES  Graphic Method 1. Draw a line 2x – y = 4. What is the solution of that linear equation? 2. Also draw a line 2x + 3y = 12. What is the solution of that linear equation?
17. 17. LINEAR EQUATION SYTEM WITH 2 VARIABLES  So, can you guess the solution of that linear equation system?  Conclusion: The solution of that linear system is the point of intersection of the lines
18. 18.  There are 4 ways to solve linear equation system with 2 variables :  Substitution  Elimination  Elimination-substitution  graph method
19. 19. - Using Substitution Method – 1. Write one of the equation in the form 2. Substitute y (or x) obtained in the first step into the other equations 3. Solve the equations to obtain the value 4. Substitute the value x=x1 obtained to get y1 or substitute the value y=y1 obtained to get x1 5. The solution set is ory ax b x cy d    1 1orx x y y    1 1,x y
20. 20.  Using Elimination Method –  The procedure for eliminating variable x (or y) 1. Consider the coefficient of x (or y). If they have same sign, subtract the equation (1) from equation (2), if they have different sign add them. 2. If the coefficients are different, make them same by multiplying each of the equations with the corresponding constants, then do the addition or subtraction as the first step. 1 1 1 2 2 2 ............. (1) ............. (2) a x b y c a x b y c    
21. 21.  Using Graphing Method – The solution of the linear system with two variables is the point of intersection of the lines.  If 2 lines are drawn in the same coordinate there are 3 possibilities of solution :  The lines will intersect at exactly one point if the gradients of the lines are different. Then the solution is unique  The lines are parallel if the gardients of the line are same. Thus, there are no solution  The line are coincide to each other if one line is a multiple of each other. Thus the solution are infinite
22. 22. SOLVING LINEAR EQUATIONS
23. 23. 1-STEP EQUATIONS: + AND – In elementary school, you solved equations that looked like this: In Algebra, we just use a variable instead of the box! We could rewrite the same equation like this: x + 3 = 5
24. 24. TO SOLVE: Need to get x by itself. We must “undo” whatever has been done to the variable. Do the same to BOTH SIDES of the = sign! Answer should be: x = some number.
25. 25. EQUATIONS HAVE TO BALANCE! Both sides of the = sign must “weigh” the same! When you change one side, do the same to the other side, so it stays balanced! x + 3 = 7
26. 26. EXAMPLES: Solve each equation. Check your answers! x + 3 = -7 m – 6 = 8
27. 27. 4 + a = -3 -8 + y = 7
28. 28. 24 = 12 + t -17 = x – 2
29. 29. YOU TRY! Solve each equation. Check your answers! x – 9 = -6 16 + n = 10
30. 30. 1-STEP EQUATIONS: × AND ÷ Just like with adding and subtracting, we need to “undo” what is done to x. Use division to “undo” multiplication. Use multiplication to “undo” division. Remember: A fraction bar means divide! Example: says x divided by 3!
31. 31. EXAMPLES: Solve each equation: 2x = 8 12 = -3b
32. 32. YOU TRY! Solve each 1-step equation. -4a = 16
33. 33. BASIC 2-STEP EQUATIONS A lot like 1-step equations! Goal: Get x by itself! Need to “undo” what is done to x. Always do the same thing to BOTH SIDES! Use Order of Operations backwards:  Add/Subtract first  Multiply/Divide last
34. 34. EXAMPLES: 4x + 7 = 27 – 6 – 2y = 10
35. 35. 5 = 7r - 2 -4 = 2 - m
36. 36. YOU TRY! Solve each equation. Check your answers! -3x + 7 = 16 -8 = 2a + 2
37. 37. MULTI-STEP EQUATIONS Multi-step equations have ( ). To solve:  Distribute first!!  Then, solve like a 2-step equation! Example: 4(-x + 4) = 12
38. 38. EXAMPLES: 3(– 6 + 3x) = 18 30 = -5(6n + 6)
39. 39. 10 (1 + 3b) = -20 -2 = -(n – 8)
40. 40. YOUR TRY! -6(1 – 5v) = 54
41. 41. 2 VARIABLES- SAME SIDE Some equations have variables in more than one place. When variables are on the same side of the = sign:  First, Combine Like Terms Like terms may NOT cross the = sign!  Then, solve like a 2-step
42. 42. EXAMPLES: 2x + 3x + 1 = 36
43. 43. 3 + 6x – 2x = -5
44. 44. -26 = -7x + 6 + 3x
45. 45. YOU TRY! Solve for x. 8x – 3x + 7 = 17
46. 46. VARIABLES ON BOTH SIDES To solve equations like this: 2x – 4 = 2 + 5x 1. Get variables together on the left. Add or subtract right side x term from both sides. 2. Solve like a 2-step equation.
47. 47. EXAMPLES 9 – 7x = -5x – 7
48. 48. -3x – 7 = 2x – 13
49. 49. YOU TRY! -3x + 9 = 44 + 4x
50. 50. UNDERSTANDING BASIC TERMS LINEAR EQUATION- Straight line or degree 1 EQUATION- Any mathematical expression equating to 0. VARIABLE- Any alphabetical constants are called variable.
51. 51. Linear Equation In one variable In two variable
52. 52. SOLUTION OF LINEAR EQUATION Solution of Linear Equation Graphical Method Algebric Method
53. 53. REPRESENTATION OF LINEAR EQUATION IN TWO VARIABLES Representation of linear Equation in two variable In algebric form In word Problem X in terms of y Y in terms of x
54. 54. No of solution in a linear Equation In one Variable It contains only one solution for Variable. In two variables It contain many or infinite number of solution for variables.