Give a geometric description of the following systems of equ

Give a geometric description of the following systems of equations. 3x - 5y = 9 4x - 3y = 9 19x - 17y = 47 4x + 20y = 0 -3x - 15y = 0 -6x - 30y = 0 3x - 5y = 9 4x - 3y = 9 19x - 17y = 45

Solution

1)

the first set of equations are

3x -5y =9

4x -3y =9

19x -17y =47

the slope of line ax+bx =c is -a/b

so finding the slope of

3x -5y =9 m(slope) = 3/5

4x -3y =9 slope = 4/3

19x -17y =47 slope 19/17

these are not identical since the lines are not same.

these are not parallel since slopes are not equal

finding the intersecting point

4(3x - 5y =9)

3(4x -3y =9)

-20y +9y =36 -27

-11y =9

y =-9/11

now plug in equation 1

3x -5(-9/11) =9

3x= 9 -45/11

3x= 54/11

x=18/11

x=18/11

now plug x=18/11 , y=-9/11 in equation3

19(18/11) -17(-9/11) =47

342/11 +153/2 =47

495/2 =47

45 =47

this is not true

so these lines satisfy

three non parallel lines with no intersecting point.-->ANSWER

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2).

4x +20y =0 slope = -1/5

-3x -15y =0 slope = -1/5

-6x-30y=0 slope = -1/5

all the slopes are equal with no constant term . that means they are identical lines

4(x +5y) =0

-3(x +5y) =0

-6(x+5y)=0   

three identical lines-->ANSWER

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3). set of lines

3x- 5y =9 slope =3/5

4x-3y=9 slope =4/3

19x-17y=45

finding intersection point between 2 lines

4(3x - 5y =9)

3(4x -3y =9)

-20y +9y =36 -27

-11y =9

y =-9/11

now plug in equation 1

3x -5(-9/11) =9

3x= 9 -45/11

3x= 54/11

x=18/11

x=18/11

now plug x=18/11 , y=-9/11 in equation3

19(18/11) -17(-9/11) =45

342/11 +153/2 =45

495/2 =45

45 =45

true

so three lines intersecting at one point (18/11 , -9/11)