Systems of equations and solution methods

Systems of Equations

Solve for X and Y

X + Y=6, 2X + Y = 8

7X + 3Y = 14, 5X + 9Y = 10

4X + Y = 16, 2X + 3Y = 24

Y = 25, 8X - 2Y = 14

Using the elimination method for solving simultaneous equations we get

X + Y=6, 2X + Y = 8

X + Y = 6 -- ( [1]

2X + Y = 8 -- ( [2]

( (Changing signs)

Substituting X = 2 in [1] we get

+ y = 6

y = 6 -- 2

y = 4

Therefore x = 2 and y = 4

X and y values thus found can be verified by substituting them in any one of the given equations.

2X + Y = 8

L.H.S = R.H.S

7X + 3Y = 14, 5X + 9Y = 10

7X + 3Y = 14 -- ((1)

5X + 9Y = 10 -- >(2)

Multiplying (1) by 5, we get

35x + 15y = 70 -- ( (3)

Multiplying (2) by 7, we get

35x + 63y = 70 -- ( (4)

(3)- (4),we get

-48y = 0

y = 0

Substituting y = 0 in [1] we get

7x + 0 = 14

=>7x = 14

Therefore x = 2 and y = 0

X and y values thus found can be verified by substituting them in any one of the given equations.

5x + 9y = 10

5(2) + 9(0) = 10

10 + 0 = 10

L.H.S = R.H.S

c. 4X + Y = 16, 2X + 3Y = 24

4X + Y = 16 -- ((1)

2X + 3Y = 24 -- ((2)

Multiplying (1) by 3, we get

12X + 3Y = 48 -- ((3)

(3)-(2), we get

10x = 24

=> x = 2.4

Substituting x = 2.4 in (1) we get

4(2.4) + y = 16

9.6 + y = 16

=> y = 16 -- 9.6

=> y = 6.4

The values are

X = 2.4, y = 6.4

Verification:-

Substituting x = 2.4,y = 6.4 in (2) we get

2(2.4) + 3(6.4) = 24

4.8 + 19.2 = 24

L.H.S = R.H.S

d. 12X + Y = 25, 8X - 2Y = 14

12X + Y = 25 -- ((1)

8X - 2Y = 14 -- ((2)

Multiplying (1) by 2, we get

24x + 2y = 50 -- ((3)

Adding (2) and (3), we get

32x = 64

Substituting x = 2 in (1) we get

12(2) + y = 25

24 + y = 25

=>y = 25 -- 24

=>y = 1

The values are

X = 2, y = 1

Verification:-

Substituting x = 2,y = 1 in (2) we get

8(2) -- 2(1) = 14

=>16 -- 2 = 14

L.H.S = R.H.S

2)

Let x be the value of a share of Company X.

Let y be the value of a share of Company Y.

The equations for Bob and Frank's holdings are

8000x + 6000y = 680000 -- ((1)

2000x + 10000y = 680000 -- ((2)

In order to simplify the given equations we divide the above two equations by 1000

8x + 6y = 680 -- ((3)

2x + 10y = 680 -- ((4)

Multiplying (4) by 4, we get

8x + 40y = 2720 -- ((5)

(3)-(5), we get

-34y = -2040

=> y = 60

Substituting y = 60 in (3) we get

8x + 6(60) = 680

8x + 360 = 680

8x = 320

The values are x = 40, y = 60

Verification:-

Substituting x = 40,y = 60 in (4) we get

2(40) + 10(60) = 680

80 + 600 = 680

L.H.S = R.H.S

The value of a share of Company X is 40

The value of a share of Company Y is 60

3)

a. X + 2Y + Z = 6 -- ((1)

X + Y = 4 -- ((2)

3X + Y + Z = 8 -- ((3)

(1)-(3), we get

-2X + Y = -2 -- ( (4)

(4)-(2), we get

-3X = -6

Substituting X = 2 in (2) we get

2 + Y = 4

=> Y = 2

Substituting X = 2, Y = 2 in (1) we get

2 + 2(2)+ Z = 6

=> 2 + 4 + Z = 6

=> Z = 0

The values are

X = 2, Y = 2,Z = 0

Verification:-

Substituting X = 2,Y = 2,Z = 0 in (3) we get

3(2) + 2 + 0 = 8

6 + 2 = 8

L.H.S = R.H.S

b. 10X + Y + Z = 12 -- ((1)

8X + 2Y +Z = 11 -- ((2)

20X - 10y - 2Z = 8 -- ((3)

(1)-(2),we get

2X -- Y = 1 -- ((4)

Multiplying (1) by 2, we get

20X + 2Y + 2Z = 24 -- ((5)

(5)+(3), we get

40X -- 8Y = 32 -- ((6)

Multiplying (4) by 8, we get

16X -- 8Y = 8 -- ((7)

(7)-(6), we get

-24X = -24

Substituting X = 1 in (4) we get

2- Y = 1

Y = 1

Substituting X = 1, Y = 1 in (1) we get

10 + 1 + Z = 12

11 + Z = 12

Z = 1

The values are

X = 1, Y = 1,Z = 1

Verification:-

Substituting X = 1,Y = 1,Z = 1 in (3) we get