Pair of linear equations in 2 variables
- 1. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Class 10
- 2. Introduction • A common form of linear equation in the two variables x and y is y=ax+c where a and c are constants. • In this equation, the a determines slope and c determines point at which line crosses the y axis. In this, (3,2) is the solution of pair of linear equations as graph for each equation cuts at point (3,2)
- 3. About Linear Equations • Linear Equation - An equation of the form ax+ by+ c= 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. • A linear equation in two variables has infinitely many solutions. • The graph of every linear equation in two variables is a straight line. • The graph of x = a is a straight line parallel to the y-axis. • The graph of y = a is a straight line parallel to the x-axis. • Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation.
- 4. Elimination method • Step 1 : First multiply both the equations by some suitable nonzero constants to make the coefficients of one variable (either x or y) numerically equal. • Step 2 :Then add or subtract one equation from the other so that one variable gets eliminated. If you get an equation in one variable, go to • Step 3. If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions. If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent. Step 3 : Solve the equation in one variable (x or y) so obtained to get its value. • Step 4 : Substitute this value of x(or y) in either of the original equations to get the value of the other variable.
- 6. Substitution Method • Step 1 : Find the value of one variable, say yin terms of the other variable, i.e., x from either equation, whichever is convenient. • Step 2 : Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved. Sometimes, as in Examples , you can get statements with no variable. If this statement is true, you can conclude that the pair of linear equations has infinitely many solutions. If the statement is false, then the pair of linear equations is inconsistent. • Step 3 :Substitute the value of x(or y) obtained in Step 2 in the equation used in Step 1 to obtain the value of the other variable.
- 8. Cross- Multiplicatio n method • The formula for cross-multiplication and its use in solving two simultaneous equations can be presented as: If (a1b2 - a2b1) ≠ 0 from the two simultaneous linear equations a1x + b1y + c1 = 0 ----------- (i) a2x + b2y + c2 = 0 ----------- (ii) We get, by the cross-multiplication method: x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)
- 10. Word Problems • Word problems based on linear equations in 2 variables can easily be solved by the following this method. • One number is 28 more than three times another number. If each number were multiplied by four, their difference would be 232.What are the numbers? Solution: a =3b +28 4a - 4b = 232 a = ? b = ? 4a - 4b = 232 4(3b +28) - 4b = 232 12b + 112 - 4b = 232 8b + 112 = 232 8b = 232 - 112 8b = 120 b = 120/8 = 15 =>b = 15 a = 3b +28 = 3(15) + 28 = 45 + 28 = 73 => a =73
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