Systems of Linear Equations Assessment, Coursework Example

Answer the question and solve the problems below.  Make sure you show all your work so you can get partial credit even if you get the final answer wrong.

1.  What is a system of equations?

A system of linear equations are two more equations that involve the same set of variables.

2. Solve for X and Y in the following problems.  Make sure you show all your work so you can get partial credit even if your final answer is wrong.

a.   X + Y= 10 ,  3X + Y = 12

Eliminating Y

-X-Y=-10

+ 3X + Y = 12

____________

2X=2

X = 1

Substituting X=1 to obtain Y

1 + Y = 10

Y = 9

Answer: X=1, Y=9

b.  2X + 5Y = 19 ,  3X + 3Y = 15

Simplifying 3X + 3Y = 15 by multiplying the equation by 1/3

X + Y = 5

Eliminating X

2X + 5Y = 19

+ -2X – 2Y = -10

________________

3Y = 9

Y = 3

Substituting to solve for X

X + 3 = 5

X = 2

Answer: X=2, Y=3

c.  4X +  Y = 22 ,    2X + 3Y = 16

Expressing Y in terms of X using the first equation

Y = 22 – 4X

Substituting this to equation 2 and solving for X.

2X + 3 (22 – 4X) = 16

2X + 66 – 12X = 16

-10X = -50

X = 5

Substituting to obtain Y

Y = 22 – 4(5)

Y = 22 – 20

Y = 2

Answer: X = 5 , Y = 2

d.  12X + Y = 174 ,  8X – 2Y = 36

Expressing Y in terms of X using the first equation

Y = 174 – 12X

Substituting this to equation 2 and solving for X.

8X – 2(174 – 12X) = 36

8X – 348 + 24X = 36

32X = 384

X = 12

Substituting to obtain Y

Y = 174 – 12 (12)

Y = 174 – 144

Y = 30

Answer: X = 12, Y = 30

2. 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 $372,000.

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 $400,000.

a.  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 = 372000

8000X + 6000Y = 400000

b. Solve for the value of a share of Company X and Company Y.  Show your work so you can get partial credit even if your final answer is wrong.

Eliminating X

-8000X – 40000Y = -1488000

+  8000X + 6000Y  = 400000

-34000Y = -1088000

Y = 32

Substituting to solve for X

2000X + 10000 (32) = 372000

2000X + 320000 = 372000

2000X = 372000 – 320000

2000X = 52000

X = 26

Answer: A share of company X is $26 and a share of company Y is $32.

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

a. X + 2Y +  Z = 22

X + Y  = 15

3X + Y + Z =  37

Eliminating Z using equations 1 and 3

-X – 2Y – Z = -22

+) 3X + Y + Z = 37

________________

2X – Y = 15

Using the equation above and equation 2 to eliminate Y and solve for X

2X – Y = 15

+)   X + Y  = 15

________________

3X = 30

X = 10

Substituting X=10 to solve for Y

10 + Y = 15

Y = 5

Substituting X = 10 and Y = 5 to solve for Z

10 + 2(5) + Z = 22

20 + Z = 22

Z = 2

Answer: X= 10, Y=5, Z=2

b. 10X + Y + Z = 603

8X + 2Y + Z  = 603

20X – 10Y – 2Z = -6

Eliminating Z using eq 1 and 2

10X + Y + Z = 603

 +) -8X – 2Y – Z  = -603

2X – Y = 0 or 2X = Y

Eliminating Z using eq 1 and ½ * eq3

10X + Y + Z = 603

+)  10X – 5Y – Z = -3

20X – 4Y = 600

(simplifying above by multiplying ¼ to the equation)

5X – Y = 150

Solving for X by substituting 2X = Y

5X – 2X = 150

3X = 150

X = 50

Solving for Y

2 (50) = Y

Y = 100

Solving for Z

10(50) + 100 + Z = 603

Z = 603 – 600

Z = 3

Answer: X = 50, Y = 100, Z = 3

c. 22X + 5Y + 7Z = 12

10X + 3Y + 2Z = 5

9X  +  2Y  + 12Z = 14

Eliminating Z using eq 2 and eq 3

-60X – 18Y – 12Z = -30

 +) 9X  +  2Y  + 12Z = 14

-51X – 16Y = -16

Eliminating Z using eq1 and eq2

44X + 10Y + 14Z = 24

+)-70X – 21Y – 14Z = -35

-26X – 11Y = -11

Eliminating Y to solve for X

-51(11)X – 16(11)Y = -16(11)

+) 26(16)X + 11(16)Y = 11(16)

(-51(11)+26(16))X = 0

X = 0

Solving for Y

-26(0) + -11Y = -11

Y = 1

Solving for Z

10(0) + 3(1) + 2Z = 5

3 + 2Z = 5

2Z = 2

Z = 1

Answer: X=0, Y = 1, Z=1

Assignment Expectations

• Define a system of equations.
• Solve systems of equations with two and three variables.