Question 64589
QUESTION:


A system of equations is a collection of two or more equations with the same set of unknowns. 


ANSWER:



In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. 



A system of linear equations can be solved four different ways: 


Substitution,

Elimination,

Matrices,

Graphing. 


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Let's discuss these methods with an example,



1.  Solve these equations.



x-y = 4____________________________(1)



3x + 2y = 7___________________________(2)



METHOD OF SUBSTITUTION:



(1)==> x = 4 + y ___________________________(3)



Substitute this value of x in (2)




==> 3(4+ y) + 2y = 7



==> 12 + 3y + 2y = 7



==> 12 + 5y = 7




==> 5y = 7 - 12




==> 5y = -5




==> y = -5/5



==> y = -1



Now put y = -1 in (3)



==> x = 4 + -1



==> x = 3



So Solution is x = 3 and y = -1



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METHOD OF ELIMINATION:



x-y = 4____________________________(1)



3x + 2y = 7___________________________(2)




Multiply equation(1) by '2' so that  coefficient of y in both equations become same( but opposite sign).  Also we could make coefficient of x same.




==> 2x-2y = 8____________________________(3)




3x + 2y = 7___________________________(2)



Now add  (3) and (2)



==> 2x-2y + 3x + 2y = 8 + 7



==> 5x = 15




==> x = 15/5



==> x = 3




Now substitute value of x=3 in (1)



==> 3-y = 4




==> 3- 4 = y



==> -1 = y




That is  x= 3 and y = -1



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METHOD OG GRAPHING.


x-y = 4____________________________(1)



3x + 2y = 7___________________________(2)



First of all rewrite the equation in terms of 'x'



(1)==> y = x-4




Now give arbitrary values to 'x' like 0,1,2,3,...-1, -2,....(atleast 5 values)

and find out corresponding values of y.



for example, if we put x = 0, y = 0-4 = -4


if x = 1, y = 1-4= -3......and so on




Now plot these ordered pairs....




Similarly,



(2)==> 2y = 7 - 3x


==> y = 7/2 - (3/2)x




Here also give values to x and find out corresponding values of y and then plot the graph.



The resulting graph will come like this,


{{{ graph( 500, 500, -10, 10, -10, 10, x-4, 7/2 - (3/2)x) }}} 



We can see that  two straight lines intersect at a single point.




Find out the coordinates of this point.




For that draw lines perpendicular to x axis and Y axis.




The perpendicular lines will meet the X axis at 3 and in the Y axis at -1


Which gives the soluton of these equations, that is,




x = 3 and y = -1





Hope you understood.





Regards.



Praseena.