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Solve the system of equations by addition.
$$\left(10,\mathrm{-4}\right)$$
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 $\text{\hspace{0.17em}}y$ -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as $\text{\hspace{0.17em}}12=0.$
Solve the following system of equations.
We can approach this problem in two ways. Because one equation is already solved for $\text{\hspace{0.17em}}x,$ the most obvious step is to use substitution.
Clearly, this statement is a contradiction because $\text{\hspace{0.17em}}9\ne 13.\text{\hspace{0.17em}}$ 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.
We then convert the second equation expressed to slope-intercept form.
Comparing the equations, we see that they have the same slope but different y -intercepts. Therefore, the lines are parallel and do not intersect.
Solve the following system of equations in two variables.
No solution. It is an inconsistent system.
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 $\text{\hspace{0.17em}}0=0.$
Find a solution to the system of equations using the addition method .
With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let’s focus on eliminating $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If we multiply both sides of the first equation by $\text{\hspace{0.17em}}\mathrm{-3},$ then we will be able to eliminate the $\text{\hspace{0.17em}}x$ -variable.
Now add the equations.
We can see that there will be an infinite number of solutions that satisfy both equations.
Solve the following system of equations in two variables.
The system is dependent so there are infinite solutions of the form $\text{\hspace{0.17em}}(x,2x+5).$
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 $\text{\hspace{0.17em}}R=xp,$ where $\text{\hspace{0.17em}}x=$ quantity and $\text{\hspace{0.17em}}p=$ price. The revenue function is shown in orange in [link] .
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