<< Chapter < Page Chapter >> Page >

Practice set a

Solve each system by addition.

{ x + y = 6 2 x y = 0

( 2 , 4 )

Got questions? Get instant answers now!

{ x + 6 y = 8 x 2 y = 0

( 4 , 2 )

Got questions? Get instant answers now!

Sample set b

Solve the following systems using the addition method.

Solve { 6 a 5 b = 14 ( 1 ) 2 a + 2 b = 10 ( 2 )

Step 1: The equations are already in the proper form, a x + b y = c .

Step 2: If we multiply equation (2) by —3, the coefficients of a will be opposites and become 0 upon addition, thus eliminating a .

       { 6 a 5 b = 14 3 ( 2 a + 2 b ) = 3 ( 10 ) { 6 a 5 b = 14 6 a 6 b = 30

Step 3:  Add the equations.

       6 a 5 b = 14 6 a 6 b = 30 0 11 b = 44

Step 4:  Solve the equation 11 b = 44.

       11 b = 44
        b = 4

Step 5:  Substitute b = 4 into either of the original equations. We will use equation 2.

       2 a + 2 b = 10 2 a + 2 ( 4 ) = 10 Solve for  a . 2 a 8 = 10 2 a = 2 a = 1

 We now have a = 1 and b = 4.

Step 6:  Substitute a = 1 and b = 4 into both the original equations for a check.

       ( 1 ) 6 a 5 b = 14 ( 2 ) 2 a + 2 b = 10 6 ( 1 ) 5 ( 4 ) = 14 Is this correct? 2 ( 1 ) + 2 ( 4 ) = 10 Is this correct? 6 + 20 = 14 Is this correct? 2 8 = 10 Is this correct? 14 = 14 Yes, this is correct . 10 = 10 Yes, this is correct .

Step 7:  The solution is ( 1 , 4 ) .

Got questions? Get instant answers now!

Solve  { 3 x + 2 y = 4 4 x = 5 y + 10 ( 1 ) ( 2 )

Step 1:  Rewrite the system in the proper form.

       { 3 x + 2 y = 4 4 x 5 y = 10 ( 1 ) ( 2 )

Step 2:  Since the coefficients of y already have opposite signs, we will eliminate y .
     Multiply equation (1) by 5, the coefficient of y in equation 2.
     Multiply equation (2) by 2, the coefficient of y in equation 1.

       { 5 ( 3 x + 2 y ) = 5 ( 4 ) 2 ( 4 x 5 y ) = 2 ( 10 ) { 15 x + 10 y = 20 8 x 10 y = 20

Step 3:  Add the equations.

       15 x + 10 y = 20 8 x 10 y = 20 23 x + 0 = 0

Step 4:  Solve the equation 23 x = 0

       23 x = 0

       x = 0

Step 5:  Substitute x = 0 into either of the original equations. We will use equation 1.

       3 x + 2 y = 4 3 ( 0 ) + 2 y = 4 Solve for  y . 0 + 2 y = 4 y = 2

 We now have x = 0 and y = 2.

Step 6:  Substitution will show that these values check.

Step 7:  The solution is ( 0 , 2 ) .

Got questions? Get instant answers now!

Practice set b

Solve each of the following systems using the addition method.

{ 3 x + y = 1 5 x + y = 3

( 1 , 2 )

Got questions? Get instant answers now!

{ x + 4 y = 1 x 2 y = 5

( 3 , 1 )

Got questions? Get instant answers now!

{ 2 x + 3 y = 10 x + 2 y = 2

( 2 , 2 )

Got questions? Get instant answers now!

{ 5 x 3 y = 1 8 x 6 y = 4

( 1 , 2 )

Got questions? Get instant answers now!

{ 3 x 5 y = 9 4 x + 8 y = 12

( 3 , 0 )

Got questions? Get instant answers now!

Addition and parallel or coincident lines

When the lines of a system are parallel or coincident, the method of elimination produces results identical to that of the method of elimination by substitution.

Addition and parallel lines

If computations eliminate all variables and produce a contradiction, the two lines of the system are parallel and the system is called inconsistent.

Addition and coincident lines

If computations eliminate all variables and produce an identity, the two lines of the system are coincident and the system is called dependent.

Sample set c

Solve { 2 x y = 1 ( 1 ) 4 x 2 y = 4 ( 2 )

Step 1: The equations are in the proper form.

Step 2: We can eliminate x by multiplying equation (1) by –2.

       { 2 ( 2 x y ) = 2 ( 1 ) 4 x 2 y = 4 { 4 x + 2 y = 2 4 x 2 y = 4

Step 3:  Add the equations.

       4 x + 2 y = 2 4 x 2 y = 4 0 + 0 = 2 0 = 2

 This is false and is therefore a contradiction. The lines of this system are parallel.  This system is inconsistent.

Got questions? Get instant answers now!

Solve  { 4 x + 8 y = 8 ( 1 ) 3 x + 6 y = 6 ( 2 )

Step 1:  The equations are in the proper form.

Step 2:  We can eliminate x by multiplying equation (1) by –3 and equation (2) by 4.

       { 3 ( 4 x + 8 y ) = 3 ( 8 ) 4 ( 3 x + 6 y ) = 4 ( 6 ) { 12 x 24 y = 24 12 x + 24 y = 24

Step 3:  Add the equations.

       12 x 24 y = 24 12 x + 24 y = 24 0 + 0 = 0 0 = 0

 This is true and is an identity. The lines of this system are coincident.

 This system is dependent.

Got questions? Get instant answers now!

Practice set c

Solve each of the following systems using the addition method.

{ x + 2 y = 6 6 x + 12 y = 1

inconsistent

Got questions? Get instant answers now!

{ 4 x 28 y = 4 x 7 y = 1

dependent

Got questions? Get instant answers now!

Exercises

For the following problems, solve the systems using elimination by addition.

{ x + y = 11 x y = 1

( 5 , 6 )

Got questions? Get instant answers now!

{ x + 3 y = 13 x 3 y = 11

Got questions? Get instant answers now!

{ 3 x 5 y = 4 4 x + 5 y = 2

( 2 , 2 )

Got questions? Get instant answers now!

{ 2 x 7 y = 1 5 x + 7 y = 22

Got questions? Get instant answers now!

{ 3 x + 4 y = 24 3 x 7 y = 42

( 0 , 6 )

Got questions? Get instant answers now!

{ 8 x + 5 y = 3 9 x 5 y = 71

Got questions? Get instant answers now!

{ x + 2 y = 6 x + 3 y = 4

( 2 , 2 )

Got questions? Get instant answers now!

{ 4 x + y = 0 3 x + y = 0

Got questions? Get instant answers now!

{ x + y = 4 x y = 4

dependent

Got questions? Get instant answers now!

{ 2 x 3 y = 6 2 x + 3 y = 6

Got questions? Get instant answers now!

{ 3 x + 4 y = 7 x + 5 y = 6

( 1 , 1 )

Got questions? Get instant answers now!

{ 4 x 2 y = 2 7 x + 4 y = 26

Got questions? Get instant answers now!

{ 3 x + y = 4 5 x 2 y = 14

( 2 , 2 )

Got questions? Get instant answers now!

{ 5 x 3 y = 20 x + 6 y = 4

Got questions? Get instant answers now!

{ 6 x + 2 y = 18 x + 5 y = 19

( 4 , 3 )

Got questions? Get instant answers now!

{ x 11 y = 17 2 x 22 y = 4

Got questions? Get instant answers now!

{ 2 x + 3 y = 20 3 x + 2 y = 15

( 1 , 6 )

Got questions? Get instant answers now!

{ 5 x + 2 y = 4 3 x 5 y = 10

Got questions? Get instant answers now!

{ 3 x 4 y = 2 9 x 12 y = 6

dependent

Got questions? Get instant answers now!

{ 3 x 5 y = 28 4 x 2 y = 20

Got questions? Get instant answers now!

{ 6 x 3 y = 3 10 x 7 y = 3

( 1 , 1 )

Got questions? Get instant answers now!

{ 4 x + 12 y = 0 8 x + 16 y = 0

Got questions? Get instant answers now!

{ 3 x + y = 1 12 x + 4 y = 6

inconsistent

Got questions? Get instant answers now!

{ 8 x + 5 y = 23 3 x 3 y = 12

Got questions? Get instant answers now!

{ 2 x + 8 y = 10 3 x + 12 y = 15

dependent

Got questions? Get instant answers now!

{ 4 x + 6 y = 8 6 x + 8 y = 12

Got questions? Get instant answers now!

{ 10 x + 2 y = 2 15 x 3 y = 3

inconsistent

Got questions? Get instant answers now!

{ x + 3 4 y = 1 2 3 5 x + y = 7 5

Got questions? Get instant answers now!

{ x + 1 3 y = 4 3 x + 1 6 y = 2 3

( 0 , 4 )

Got questions? Get instant answers now!

{ 8 x 3 y = 25 4 x 5 y = 5

Got questions? Get instant answers now!

{ 10 x 4 y = 72 9 x + 5 y = 39

( 258 7 , 519 7 )

Got questions? Get instant answers now!

{ 12 x + 16 y = 36 10 x + 12 y = 30

Got questions? Get instant answers now!

{ 25 x 32 y = 14 50 x + 64 y = 28

dependent

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify and write ( 2 x 3 y 4 ) 5 ( 2 x y 6 ) 5 so that only positive exponents appear.

Got questions? Get instant answers now!

( [link] ) Simplify 8 + 3 50 .

17 2

Got questions? Get instant answers now!

( [link] ) Solve the radical equation 2 x + 3 + 5 = 8.

Got questions? Get instant answers now!

( [link] ) Solve by graphing { x + y = 4 3 x y = 0
An xy coordinate plane with gridlines labeled negative five and five with increments of one unit for both axes.

( 1 , 3 )
A graph of two lines intersecting at a point with coordinates negative one, three. One of the lines is passing through a point with coordinates zero, zero and the other line is passing through two points with coordinates zero, four and four, zero.

Got questions? Get instant answers now!

( [link] ) Solve using the substitution method: { 3 x 4 y = 11 5 x + y = 3

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask