Exercises_Systems of Linear Equations

INTERMEDIATE INTERMEDIA TE ALGEBRA

Exercises on Systems of Linear Equations  Direction:  Solve

the following. Show all necessary computations.

1. State whether the given given system system is inconsisten inconsistent, t, consisten consistent-depen t-dependen dent, t, or consistent consistent-independent. (a)

�√  − √  √  � √  �− −

2x 3y  =  π √  7 2x + π 2 = 7 3y

(1 + 2)4 x  = 7π y − π7 √  16(7π )y  = −(2 + 16)4 x − 16π 7 3

(b) (c)

3

2x 5y  = 1 3x + 2y  = 2

2. Use the substitution substitution and elimination method, and Cramer’s Cramer’s rule to find all solutions of each each system. system. Graph Graph each each system system to chec check k your answ answer. er. (If  D   = 0, use other methods.) (a) (b) (c) (d) (e)

�− � −− �− �− −− � − 2 5

1 4

x

x + 41 y  = 3 2 5

y  =

−3

13x 8y  = −3 7x + 2 y  = 0 4x + 6y  = 3 6x 9y  = − 92 3 2

5y  = 1 x + 43 y  = −1 x

6x 2y  = −3 5x + 3y  = 4

3. Solve Solve the given given system system using using Cramer Cramer’s ’s rule. If  D  = 0, use the pairwise-elimination method.

(a)

 

7x + 5 y − 7z  = −10 2x + y  +  z  = 7 x + y − 3z  = −8

(b)

(c)

  −−  − −  −  

3x + y − z  = 10 8x y − 6z  = −3 5x 2y − 5z  = 1 2x y +  z  = −1 x + 3 y − 2z  = 2 5x + 6y − 5z  =

−5

x + y  +  z  = 1

(d)

(e)

2x + y  +  z  = −2 3x + 6 y + 6 z  = 5 3x + 3 y − 2z  = 13 6x + 2 y − 5z  = 13 7x + 5 y − 3z  = 26

4. Solve the following problems completely. Show your solution. (a) A certain alloy contains 30% tin and 40% copper. (The percentages are by weight.) How many pounds of tin and how many pounds of copper must be melted with 500 lb of the given alloy to yield a new alloy containing 50% tin and 60% copper? (b) The sum of the digits of a three-digit number is 17. If the hundreds and ones digits were reversed, the new number is 495 less than the original number. If  the ones and tens digits were switched, the original number decreases by 54. Find the original number. (c) An airplane travels 360 km. with the wind in 1 hr. and 20 mins., and returns against the wind in 2 hrs. and 15 mins. Find the speed of the wind. (d) The speed of a motorboat in still water is 16 mph. Find the speed of the river’s current if the motorboat goes 5 mi. downstream in the same time required going 3 mi. upstream. (e) A shopkeeper has two types of coffee beans on hand. One type sells for PhP 5.2/kg, the other for PhP 5.8/kg. How many kilograms of each type must be mixed to produce 16 kg of a blend that sells for PhP 5 .5/kg? (f) Given that the lines 7x + 5y  = 4, x + ky  = 3, and 5x + y  +  k  = 0 are concurrent (pass through a common point), what are the possible values for k? (g) Solve the following system for s and t:

�  − 1 2s 2

1 2t 3

s

t

+

= −10 =5

Hint: Let x  = 1s and y  = 1t . Solve for the resulting linear equation. After solving for x  and y , use the fact that s  = x1 and t  = y1 .

(h) Solve the following system for r , s, and t:

  − −  − � 2

r

6

r

3

+ 5t = 15 2 + 1s + 4t = 0 r + 2s 9t = −12 s

(i) Solve the following system for x  and  y : 2x2 + 2y 2 = 55 4x2 − 8y 2 = 109

(j) What should be the value of  m  so that the system solution?

� �

� �

�

mx + y  = 3

6y − 3x  = 11

has no

1 2 5 6 5. Let  A  = and  B  = . Let  A 2 and  B 2 denote the matrix products  AA  and  B B , 3 4 7 8 respectively. Compute each of the following. (a) (A + B )(A + B ) (b) A2 + 2AB  +  B 2 (c) A2 + AB  +  BA + B 2