Solving systems of linear equations with two and three variables

System of Linear Equations

Solve for X and Y in the following problems. Make sure you show all your work.

X + Y=7, X + 2Y = 11

y

y + 2y = 11

y = 4

3X + 7Y = 24, 9X + 5Y = 24

7/3y

7/3 y) + 5y = 24

63/3y + 5y = 24

y = 3

4X + Y = 17, 2X + 3Y = 21

y = 17 -- 4x

2x + 3(17-4x) = 21

+ y = 17

y = 5

18X + Y = 42, 6X - 3Y = -6

y = 42 -- 18x

+ y = 42

y = 6

Suppose Bob owns 2,000 shares of Company X and 10,000 shares of Company Y. The total value of Bob's holdings of these two companies is $340,000.

Suppose Frank owns 8,000 shares of Company X and 6,000 shares of Company Y. The total value of Franks holdings of these two companies is $340,000.

Write equations for Bob and Frank's holdings. Use the variables X and Y to represent the values of shares of Company X and Company Y.

2000X + 10000Y = 340,000

8000X + 6000 Y = 340,000

Solve for the value of a share of Company X and Company Y. Show your work.

2000x = 340,000 -- 10000y

x = 170 -- 5y

8000 (170 -- 5y) + 6000y = 340,000

1,360,000 -- 40,000y + 6000y = 340,000

-34,000y = -1020000

y = 30

2000x + 300000 = 340,000

2000x = 40,000

3. Solve for X, Y, and Z. In the following systems of three equations:

a. X + Y + 2Z = 5

X + Y = 5

X + 3Y + Z = 11

z = 11 -- x -- 3y

x = 5 -- y

(5 -- y + y + 2(11-x-3y) = 5

5 + 22 -- 2x -- 6y = 5

27 -- 2 (5-y) -- 6y = 5

27 -- 10 + 2y -- 6y = 5

17 -- 4y = 5

-4y = -12

y = 3

z = 0

b. X + 10Y + Z = 52

5X + Y + 4Z = 15

X + 2y - 3Z = 12

-5x -- 50 y -- 5z = -260

5x + y + 4z = 15

-49y -- 1z = -245

x + 10 y + z = 52

-x -2y + 3z = -12