<< Chapter < Page Chapter >> Page >

Solve the system of equations by addition.

2 x + 3 y = 8 3 x + 5 y = 10

( 10 , −4 )

Got questions? Get instant answers now!

Identifying inconsistent systems of equations containing two variables

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system    consists of parallel lines that have the same slope but different y -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as 12 = 0.

Solving an inconsistent system of equations

Solve the following system of equations.

         x = 9 −2 y x + 2 y = 13

We can approach this problem in two ways. Because one equation is already solved for x , the most obvious step is to use substitution.

x + 2 y = 13 ( 9 2 y ) + 2 y = 13 9 + 0 y = 13 9 = 13

Clearly, this statement is a contradiction because 9 13. Therefore, the system has no solution.

The second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.

   x = 9 −2 y 2 y = x + 9    y = 1 2 x + 9 2

We then convert the second equation expressed to slope-intercept form.

x + 2 y = 13        2 y = x + 13           y = 1 2 x + 13 2

Comparing the equations, we see that they have the same slope but different y -intercepts. Therefore, the lines are parallel and do not intersect.

y = 1 2 x + 9 2 y = 1 2 x + 13 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the following system of equations in two variables.

2 y −2 x = 2 2 y −2 x = 6

No solution. It is an inconsistent system.

Got questions? Get instant answers now!

Expressing the solution of a system of dependent equations containing two variables

Recall that a dependent system    of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as 0 = 0.

Finding a solution to a dependent system of linear equations

Find a solution to the system of equations using the addition method    .

x + 3 y = 2 3 x + 9 y = 6

With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let’s focus on eliminating x . If we multiply both sides of the first equation by −3 , then we will be able to eliminate the x -variable.

              x + 3 y = 2   ( −3 ) ( x + 3 y ) = ( −3 ) ( 2 )          −3 x 9 y = 6

Now add the equations.

3 x 9 y = −6 + 3 x + 9 y = 6 ______________ 0 = 0

We can see that there will be an infinite number of solutions that satisfy both equations.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the following system of equations in two variables.

     y −2 x = 5 −3 y + 6 x = −15

The system is dependent so there are infinite solutions of the form ( x , 2 x + 5 ) .

Got questions? Get instant answers now!

Using systems of equations to investigate profits

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function    is the function used to calculate the amount of money that comes into the business. It can be represented by the equation R = x p , where x = quantity and p = price. The revenue function is shown in orange in [link] .

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, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask