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Systems of Linear Equations: Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Beginning with the graphical solution of systems, this chapter includes an interpretation of independent, inconsistent, and dependent systems and examples to illustrate the applications for these systems. The substitution method and the addition method of solving a system by elimination are explained, noting when to use each method. The five-step method is again used to illustrate the solutions of value and rate problems (coin and mixture problems), using drawings that correspond to the actual situation. This module contains the proficiency exam for the chapter "Systems of Linear Equations".

Proficiency Exam

Exercise 1

((Reference)) Solve using graphing: { 3x+2y=4 15x+10y=10 { 3x+2y=4 15x+10y=10
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

inconsistent
A graph of two parallel lines. One line is labeled with the equation three x plus two y equals four and passes through the points zero, two and two, negative one. A second line is labeled with the equation fifteen x plus ten y equals negative ten and passes through the points zero, negative one and two, negative four.

Exercise 2

((Reference)) Solve using graphing: { 2x3y=2 x+2y=8 { 2x3y=2 x+2y=8
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

( 4,2 ) ( 4,2 )
A graph of two lines intersecting at a point with coordinates four, two. One line is labeled with the equation x plus two y equals eight and passes through the points zero, four.  A second line is labeled with the equation two x minus three y equals two and passes through the points zero, negative two over three and one, zero.

Exercise 3

((Reference)) Solve using substitution: { 2x+6y=16 x4y=13 { 2x+6y=16 x4y=13

Solution

( 1,3 ) ( 1,3 )

Exercise 4

((Reference)) Solve using addition: { 3x+8y=5 x2y=3 { 3x+8y=5 x2y=3

Solution

( 1,1 ) ( 1,1 )

Exercise 5

((Reference), (Reference)) Solve using either substitution or addition: { 4x4y=8 xy=5 { 4x4y=8 xy=5

Solution

inconsistent

Exercise 6

((Reference), (Reference)) Solve using either substitution or addition: { 9x+3y=12 3xy=4 { 9x+3y=12 3xy=4

Solution

( 4 3 ,0 ) ( 4 3 ,0 )

Exercise 7

((Reference)) The sum of two numbers is 43 and the difference of the same two numbers is 7. What are the numbers?

Solution

18 and 25

Exercise 8

((Reference)) A chemist needs 80 ml of an 18% acid solution. She has two acid solutions, A and B, to mix together to form the 80-ml solution. Acid solution A is 15% acid and acid solution B is 20% acid. How much of each solution should be used?

Solution

32 ml of solution A;  48 ml of solution B. 32 ml of solution A;  48 ml of solution B.

Exercise 9

((Reference)) A parking meter contains 32 coins. If the meter contains only nickels and quarters, and the total value of the coins is $4.60 $4.60 , how many of each type of coin are there?

Solution

17 nickels and 15 quarters

Exercise 10

((Reference)) A person has $15,000 to invest. If he invests part at 8% and the rest at 12%, how much should he invest at each rate to produce the same return as if he had invested it all at 9%?

Solution

$11,250 at 8%;  $3,750 at 12% $11,250 at 8%;  $3,750 at 12%

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