Systems of Linear Equations Project, Coursework Example

A system of equation is a series of equations that have similar unknowns. To solve these equations, the most satisfying amounts are determined from the unknowns.

a) x+y=10

3x+y=12

2x=2

X=2

Y=10-x

Y=9

b) 2x+5y=19

3X + 3Y = 15

Multiply by 3 and 2

6x+15y=57

6x+6y=30

9y=27

Y=3

Using 2x+5y=19

2x+5(3) =19

2x=4

X=2

c) 4X + Y = 22

2X + 3Y = 16

Multiply by 3 and 1

12x+3y=22

2x+3y=16

10x=50

X=5

Using 4x+y=22

4(5 )+y=22

Y=2

d) 12X + Y = 174

8X – 2Y = 36

Multiply by 2 and 1

24x+2y=348

8x-2y=36

32x  = 384

X= 12

8(12)-2y=36

-2y=-60

Y=30

8,000x+6,000y=400,000

2,000x+10,000y=372,000

8x+6y=400

2x+10y=372

Multiply by 2 and 8

16x+12y=800

16x+80y=2976

-68y=-2176

Y=32

8x+32(6)=400

8x=400-192

X=26

a) X + 2Y + Z = 22

X +Y = 15

3X + Y + Z = 37

x=22-2y-z

Substitute in x+y=15

22-2y-z+y=15

Y+z=7

3(22-2y-z) +y+z=37

66-6y-3z+y+z=37

5y+2z=29

5y+2z=29

y+z=7

Multiply by 1 and 2

5y+2z=29

2y+2z=14

3y=  15

Y=5

Y+z=7

5+z=7

Z=2

X+y=15

X+ 5=1 5

X=10

b) 10X + Y + Z = 603

8X + 2Y + Z = 603

20X – 10Y – 2Z = -6

-2z=-6-20x+10y

Z=3+10x-5y

10x+y+3+10x- 5y=603

20x-4y=600

8x+2y+3+10x-5y=603

18x-3y=600

20x-4y=600

18x-3y=600

Multiply by 3 and 4

60x-12y=1800

72-12y=2400

12x   =600

X=50

20x- 4y=600

20(50)-4y=600

-4y=600-1000

Y=100

Z=3+10(50)-5(100)

= 3+500-500

Z= 3

c) 22X + 5Y + 7Z = 12

10X + 3Y + 2Z = 5

9X + 2Y + 12Z = 14

2z=5-10x-3y

z=5/2-5x-3/2y

22x+5y+7(5/2-5 x-3/2y)=12

22x+5y+35/2-35x-21/2y=12

22x+5y+17.5-35x-10.5y=12

13x+5. 5y=5.5

9x+2y+12(5/2-5-3/2y)=14

9x+2y+30-60x-18y=14

5 1x+16y=16

13x+5.5y=5.5

51x+16y+16

Multiply by 5 1 and 13

663x+280.5y=280.5

663x+208y  =208

72.5y=72.5

Y=1

51x+16(1)=16

51x=0

X=0

10x+3y+2z=5

10(0)+3y+2z=5

2z=2

Z=1