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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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