<< 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

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask