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Solve the system of equations by addition.

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

( 10 , −4 )

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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
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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.

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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.

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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 ) .

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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

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
more than 6000
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
Thanks po.
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
What is domain
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply

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