linear_systems

linear_systems - Linear Systems of Equations in two...

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Linear Systems of Equations in two variables Example 1 : Question : Solve the system of equations and interpret the solution graphically: { = y + x 2 = + x 3 y 18 . Solution : The problem is to find two numbers x and y such that both of the given equations hold simultaneously. We consider three different methods of obtaining the solution. Method I : Subsitution method. { = y + x 2 ------- ( ) 1 = + x 3 y 18 ------- ( ) 2 Substituting = y + x 2 from equation (1) in equation (2) gives: = + x 3 ( ) + x 2 18 <=> = + + x 3 x 6 18 <=> = 4 x 12 <=> = x 3 When = x 3, equation (1) gives = y 5. The two values = x 3 and = y 5 can be checked in equation (2). The solution is given by = x 3 and = y 5. Method II : First elimination method. Arrange the equations with the unknowns x and y on the left side with the x term before the y term. { = y + x 2 ------- ( ) 1 = + x 3 y 18 ------- ( ) 2 <=> { = - + x y 2 ------- ( ) 3 = + x 3 y 18 ------- ( ) 4 The variable x can be " eliminated " by adding the equations (3) and (4) together (adding the two left-hand sides together and the two right-hand sides together) to give = 4 y 20. This gives = y 5. Then x can be obtained by substituting this value in equation (1) to give = 5 + x 2, so that = x 3. Alternatively, multiplying both sides of equation (3) by 3 will mean that the y terms are the same in the two equations. <=> { = - + 3 x 3 y 6 ------- ( ) 5 = + x 3 y 18 ------- ( ) 6 Subtracting equation (5) from equation (6) ( subtracting the left-hand side of equation (5) from the left-hand side of (6) and the right-hand side of equation (5) from the right-hand side of (6) ) eliminates y to give: = 4 x 12. Hence = x 3. The solution is given by = x 3 and = y 5.
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Method III : Second elimination method. { = y + x 2 ------- ( ) 1 = + x 3 y 18 ------- ( ) 2 Arrange each of the equations in the standard form = y + m x b for the equation of a straight line with slope m and y intercept b . The first equation already has this form, and the second equation is equivalent to = 3 y - + x 18 which gives = y - 1 3 + x 6. = y + x 2 ------- ( ) 7 = y - + x 3 6 ------- ( ) 8 The variable y can be eliminated between equations (7) and (8) simply by equating the two right hand sides, to give = + x 2 - 1 3 + x 6 ------- (9) This equation is equivalent to + x 1 3 = x 4 that is, 4 3 = x 4 Hence = x 3 1 so that = x 3. Substituting this value for x in equation (7) gives = y 5. The solution is given by: = x 3 and = y 5. Graphical illustration of the solution . Consider the system in the form used for the third solution method in which both equations are arranged in the standard form = y + m x b for the equation of a straight line. = y + x 2 ------- ( ) 7 = y - + x 3 6 ------- ( ) 8 Each pair of real numbers ( ) , x y that satisfy the equation = y + x 2 corresponds a point on the straight line graph of this first equation, and similarly, the each pair of real numbers ( ) , x y that satisfy the equation = y - 1 3 + x
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This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.

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linear_systems - Linear Systems of Equations in two...

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