Question

Solve the system by graphing. (Enter your answers as a comma-separated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)

x + y = 7
−x − y = −7
(x, y)

=
Solve the system by the elimination method. (Enter your answers as a comma-separated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)

leftbrace2.gif
x + y = 12
2x + 3y = 31

Solve the system by the elimination method. (Enter your answers as a comma-separated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.)

leftbrace2.gif
3x + y = 15
x + 2y = 10

Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $4,000 or $8,000. If the partnership raised $348,000, then how many investors contributed $4,000 and how many contributed $8,000?

x = $4,000 investors
y = $8,000 investors

Answer 1

Part 1) The system has infinite solutions. Is a DEPENDENT system

Part 2) The solution of the system is the point (5,7)

Part 3) The solution of the system is the point (4,3)

Part 4) The number of investor that contributed with $4,000 was 33 and the number of investor that contributed with $8,000 was 27

Explanation:

Part 1) we have

`x+y=7` ------> equation A

`-x-y=-7` ------> equation B

Solve the system by graphing

Remember that the solution is the intersection point both graphs

using a graphing tool

The system has infinity solutions (both lines are identical)

see the attached figure

Is a DEPENDENT system

Part 2) we have

`x+y=12` ------> equation A

`2x+3y=31` ------> equation B

Solve the system by the elimination method

Multiply equation A by -2 both sides

`-2(x+y)=12(-2)`

`-2x-2y=-24` ------> equation C

Adds equation B and C and solve for y

`2x+3y=31\\-2x-2y=-24\\---------\\3y-2y=31-24\\y=7`

Find the value of x

substitute the value of y in the equation A (or B or C) and solve for x

`x+(7)=12`

`x=5`

The solution is the point (5,7)

Part 3) we have

`3x+y=15` ------> equation A

`x+2y=10` ------> equation B

Solve the system by the elimination method

Multiply equation A by -2 both sides

`-2(3x+y)=15(-2)`

`-6x-2y=-30` -----> equation C

Adds equation B and equation C

`x+2y=10\\-6x-2y=-30\\---------\\x-6x=10-30\\-5x=-20\\x=4`

Find the value of y

substitute the value of x in the equation A (or B or C) and solve for y

`3(4)+y=15`

`12+y=15`

`y=3`

therefore

The solution is the point (4,3)

Part 4) Formulate the situation as a system of two linear equations in two variables

Let

x ----> the number of investor that contributed with $4,000

y ----> the number of investor that contributed with $8,000

we have that

The system of equations is

`x+y=60` ------> equation A

`4,000x+8,000y=348,000` -----> equation B

Solve the system by elimination method

Multiply by -4,000 both sides equation A

`-4,000(x+y)=60(-4,000)`

`-4,000x-4,000y=-240,000` -----> equation C

Adds equation B and equation C and solve for y

`4,000x+8,000y=348,000\\-4,000x-4,000y=-240,000\\-----------\\8,000y-4,000y=348,000-240,000\\4,000y=108,000\\y=27`

Find the value of x

Substitute the value of x in the equation A ( or equation B or equation C) and solve for x

`x+27=60`

`x=33`

so

The solution of the system is the point (33,27)

therefore

The number of investor that contributed with $4,000 was 33 and the number of investor that contributed with $8,000 was 27