System of equations or Simultaneous equations A pair of linear equations in two variables is said to form a system of simultaneous (together) linear equations. For Example, 2x — 3y +4 = O Form a system of two linear equations in variables x and y.
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The general form of a linear equation in two variables x and y is ax + by + c = = or not O, or O, where a, b and c being real numbers. A solution of such an equation is a pair of values, one for x and the other for y, which makes two (LHS AND RI-IS) sides of the equation equal. Every linear equation in two variables has infinitely many solutions which can be represented on a certain line.
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GRAPHICAL SOLUTIONS OF A LINEAR EQUATION 'Let us consider the following system of two simultaneous linear equations in two variable. 2x-y=-1 assign any value to one of the two Here we variables and then determine the value of the other variable from the given equation.
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For the equation 2x —y = -1---(1) Solve for y ADD 1 and add y to both sides Switching sides Solve for y Subtract 3x on both sides 2y=9-3x Divide by 2 on both sides X 1 3 5
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(0,1) Х' (2,5) 3) х
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ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS 1. SUBSTITUTION METHOD 2. EQUATING OF COEFFICIENTS 3. CROSS MULTIPLICATION
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SUBSTITUTION Let the equations be + biy + = O (ii) a2X + b2b/ + = 0 STEPS 1. Choose either of the two equations, say (i) and find the value of one variable , say 'y' in terms of x 2. Substitute the value of y, obtained in the previous step in equation (ii) to get an equation in x 3. Solve the equation obtained in the previous step to get the value of x. 4. Substitute the value of x and get the value of y.
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ELIMINATION BY SUBSTITUTION Let us take an example 2x-3y= 12
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1. Choose either of the two equations, say (i) and find the value of one variable , say 'x' in terms of y Subtract 2y on both sides x = -2y -1 2. Substituting the value of x in equation 2x — 3y = 12 (ii), we get solving brackets - 4y— 2 — 3y = 12 ( rearranging - = 14 divide by -7
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Putting the value of y in eq (iii), we get x=-2y 1 -2 x (-2)-1 = 4—1 Hence the solution of the equation is (x,y)
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ELIMINATION METHOD 'We eliminate one of the two variables to obtain an equation in one variable which can easily be solved. Putting the value of this variable in any of the given equations, the value of the other variable can be obtained. 'For example: we want to solve, 3x + 2y = 11
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STEP 1 Let + 2 y = 11 - 1. Multiply 3 in equation (i) 2. Multiply 2 in equation (ii) 3. Subtracting eq iv from iii, we get = 33 - = 8 ------- (iv) = 25
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Putting the value of y in equation (ii) we get, 2x + 3y put 2 x 5 + 3y = 4 10 + 3y = 4 ( subtract by 10 both sides) 3y = 4— 10 (simplify) ( divide by 3) 3y = -6 and Y = -2 Hence,
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THANK YOU -Prepared by Mr JAY BARIA TUTOR
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