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Linear equations in 2 variables

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Linear equations in 2 variables

  1. 1. LINEAR EQUATION A Linear Equation is an algebric equation in which terms are a constants or the product of a constants and variables. Linear Equations can have one or more variables.
  2. 2. VARIABLES & CONSTANTS • VARIABLES are the unknown part of a Linear Equation , they are represented with alphabets . Like->x, y, z, a, b, c, etc. • Constants are the fixed parts of Linear Equations . The constants may be numbers, parameters, or even non- linear functions of parameters, and the distinction between variables and parameters may depend on the problem .
  3. 3. LINEAR EQUATIONS IN ONE VARIABLE The equations which can be written in the form -> ax + b = 0 , where a ≠ 0 These type equations are called Linear Equations In One Variable . Examples -> 1) 5x + 9 = 0 2) 39a – 5 = 0 3) 345u = -234 4) 5z = 0 ETC.
  4. 4. LINEAR EQUATIONS IN TWO VARIABLES The equations which can be written in the form-> ax + by – c = 0 , where a & b both can never be 0 These type equations are called Linear Equations In Two Variable . Examples -> •47x + 7y = 9 •73a – 61b = – 13 •44u + 10v – 155 = 0 •30p + 100 q = 0
  5. 5. GRAPHS OF LINEAR EQUATIONS IN ONE & TWO VARIABLES ONE VARIABLETWO VARIABLE
  6. 6. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Each linear equation in two variables defined a straight line. To solve a system of two linear equations in two variables, we graph both equations in the same coordinate system. The coordinates of any points that graphs have in common are solutions to the system, since they satisfy both equations. The general form of a pair of linear equations in two variables x and y as a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 where a1, a2, b1, b2, c1, c2 are all real numbers and a1 2 + b1 2 ≠ 0 and a2 2 + b2 2 ≠ 0.
  7. 7. METHODS FOR SOLVING PAIR OF LINEAR EQUATIONS IN TWO VARIABLES There are two methods for solving PAIR OF LINEAR EQUATIONS IN TWO VARIABLES  (1) GRAPHICAL Method (2) ALGEBRAIC Method
  8. 8. GRAPHICAL METHOD FOR SOLVING PAIR OF LINEAR EQUATIONS IN TWO VARIABLES When a pair of linear equations is plotted, two lines are defined. Now, there are two lines in a plane can intersect each other, be parallel to each other, or coincide with each other. The points where the two lines intersect are called the solutions of the pair of linear equations. Condition 1: Intersecting Lines If a1a2a1a2 ≠ b1b2b1b2 , then the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has a unique solution. Condition 2: Coincident Lines If a1a2a1a2 = b1b2b1b2 = c1c2c1c2 , then the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has infinite solutions. Condition 3: Parallel Lines If a1a2a1a2 = b1b2b1b2 ≠ c1c2c1c2 , then a pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has no solution. A pair of linear equations which has no solution is said to be an Inconsistent pair of linear equations. A pair of linear equations, which has a unique or infinite solutions are said to be a Consistent pair of linear equations.
  9. 9. GRAPHS OF ALL THREE CONDITIONS Intersecting Lines Coincident Lines
  10. 10. ALGEBRAIC METHOD FOR SOLVING PAIR OF LINEAR EQUATIONS IN TWO VARIABLES There are three Algebraic Methods for solving PAIR OF LINEAR EQUATIONS IN TWO VARIABLES  1. Elimination by Substitution Method 2. Elimination by Equating Coefficient Method 3. Cross Multiplication Method
  11. 11. (1) Elimination by Substitution Method Steps  1. The first step for solving a pair of linear equations by the substitution method is to solve one equation for either of the variables. 2. Choosing any equation & any variable for the first step does not affect the solution for the pair of equations . 3. In the next step, we’ll put the resultant value of the chosen variable obtained in the chosen equation in another equation and solve for the other variable. 4. In the last step, we can substitute the value obtained of one variable in any one equation to find the value of the other variable.
  12. 12. (2) Elimination by Equating Coefficient Method Steps  1. Equate the non-zero constants of any variable by multiplying the constants of a same variable in both equations with other equation, so that the resultant constants of one variable in both equations become equal. 2. Subtract one equation from another, to eliminate a variable and find the value of that variable 3. Solve for the remaining variable by putting the value of one solved variable .
  13. 13. (3) Cross Multiplication Method 1)) Let’s consider the general form of a pair of linear equations a1x + b1y + c1 = 0 , and a2x + b2y + c2 = 0. 2)) To solve the pair of equations for x and y using cross- multiplication, we’ll arrange the variables x and y and their coefficients a1, a2, b1 and b2, and the constants c1 and c2 as shown below 3)) Now simplifying the above situation, and putting the values of x with 1 & y with 1 to find the value of x & y x / (b1*c2-b2*c1) = y / (c1*a2- c2*a1) = 1 / (a1*b2-a2*b1)
  14. 14. (3) Cross Multiplication Method Continued These are the steps as like shown in the picture. Description on corresponding before page
  15. 15. EXAMPLES Elimination by Substitution Method
  16. 16. Elimination by Equating Coefficient Method
  17. 17. CROSS MULTIPLICATION METHOD
  18. 18. Prepared By  BHAVYAM ARORA Roll No.  37 Class 10 ‘A’ Submitted To  Mr. SATYAVEER Sir

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