<< 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 sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc 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