Solve the given linear system.

Question

Solve the given linear system.

 

a. y = 5x - 3
    y = 3x - 1  

 

b. -2x + 3y = 8
     3x + 5y = -12

 

c. 2x - 4y = -6
   -x + 2y = 3

 

d. 5x + 10y = 70
    5x + 25z = 270
    10y + 25z = 300

 

e. 2x - y - z = 5
    x + y + z = 7
    3x - 2y - 3z = 1

Answer 1

a.  Use substitution to solve   (substitute for y and set them equal to each other)

y = 5x - 3
y = 3x - 1

 

 5x - 3 = 3x - 1    

-3x       -3x

______________

2x - 3 = - 1

    +3     +3

______________

2x = 2

x = 1  

Plug in the value of x to find y

y = 5(1) - 3

y = 2

(1,2)

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b. Use elimination to solve the problem (Try to get rid of one variable I picked x to get rid of)

 (multiplying by 3 to the top Eq. & multiplying by 2 to the bottom Eq.)

 3 (  -2x + 3y = 8  )
 2 ( 3x + 5y = -12 )

_________________

 

-6x + 9y = 24

 6x +10y = -24

_________________ now x can be eliminated.

19y = 0 

y = 0

Plug in the value of y to find x

-2x + 3y = 8

-2x + 3(0) = 8

-2x  = 8

x = -4

 

(-4,0)

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c.  For this problem i will use substitution sove for x for the 2nd equation

2x - 4y = -6
-x + 2y = 3

 

-x + 2y = 3 

+x             +x

______________

2y = 3 + x 

-3    -3

______________

x = 2y - 3   Now Plug in 2y -3 in the x for the 1st equation

 

2x - 4y = -6

2 (2y - 3) - 4y = -6

4y - 6 -4y = -6

     +6         +6

0y = 0

0 = 0

 when you have a 0 = 0 that means infinite many solution for this 2 equations  ( coincided lines.)

 

infinite many solution

 

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d. I took 1st 2 equation and try to eliminate a variable x by multiplying by (-1) to 1st equation

5x + 10y = 70 
5x + 25z = 270
10y + 25z = 300

 

-1 (5x + 10y = 70)  >>>>>>    -5x - 1oy = -70
5x + 25z = 270 >>>>>>>>>    5x + 25z = 270 

-5x - 1oy = -70

5x + 25z = 270

 

_______________ now x is eliminated and new equation will be form

   -10y + 25z = 200   ( now take the 3rd equation and do elimination y.

   10y + 25z = 300

____________________

50z = 500

 

z = 10  plug in z to find y

 

10y + 25z = 300

10y + 25(10) = 300

10y + 250 = 300

       -250     -250

10y = 50

 

y = 5 plug in y to find x

 

5x + 10(5) = 70

 

5x + 50 = 70

     -50     -50

5x = 20

x = 4

 

(4, 5, 10)

 

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e.   Eliminate z & y from first 2 equation

2x - y - z = 5  
 x + y + z = 7
3x - 2y - 3z = 1

 

 

2x - y - z = 5
x + y + z = 7

__________

 

3x = 12

x = 4    Now take 2nd and 3rd equation and plug in x 

 

x + y + z = 7  >>>>>>>>>>>   4 + y + z = 7   >>>>>>>     y + z = 3
3x - 2y - 3z = 1 >>>>>>>>>>  3(4) - 2y - 3z = 1  >>>>>   - 2y - 3z = -11

 

 

    y + z = 3

- 2y - 3z = -11    Now eliminate y by mulitplying by 2 to the 1st equation

 

 2( y + z = 3 )

- 2y - 3z = -11

 

  2y + 2z = 6 
- 2y - 3z = -11

_____________

-z = -5

 

z = 5  Now plug in z and x value to find y

 

x + y + z = 7

4 + y + 5 = 7

 9 + y = 7

-9         -9

y = -2

 

(4, -2, 5)

 

Answer 2

y = 5x - 3
 y = 3x - 1

5x-3=3x-1

2x=2

x=1

y=5*1-3=2 or

y=3*1-1=2

 

-2x + 3y = 8
 3x + 5y = -12

Multiply the first equation by 5 and the second by 3 to get

-10x+15y=40

9x+15y=-36

Subtract to get

-19x=76 or x=-4, which substituted in -2x + 3y = 8 gives

8+3y=8 or y=0

 

 2x - 4y = -6
   -x + 2y = 3

Multiply the second equation by -2 and get

2x-4y=-6, the first equation.  The solution is any point on the line 2x - 4y = -6

or in standard form y=1/2x+3/2