Set up the system:

Multiply both sides of the first equation by 5, and both sides of the second by :

Add both sides of the equations, then solve for  in the resulting equation:

In the equation , substitute  for :

Simplify, then solve for  by isolating the variable:

Multiply both sides of each equation by the least common denominators of their coefficients in order to make those coefficients into whole numbers:

, so

, so

This sets up the system

Multiply the latter equation by  on both sides, then add to the former equation to eliminate the  terms:

Solve for :

In the equation , substitute  for :

Simplify the left expression:

Now solve for :

Substitute this value in the following equation and evaluate for :

One way to solve this is to know that the two equations of a linear system without a solution - an inconsistent system - are represented by a pair of parallel lines, which are lines with the same slope. To find the slope of the line of the equation , rewrite the equation in the slope-intercept form , with  the slope of the line. This is done by solving for , so isolate  in the equation as follows:

The slope of the line is , the coefficient of .

The equations in all four choices are already in slope-intercept form, so examine the coefficient of  in each. The equation  is the only one with -coefficient , so its slope is also . Since the lines of the equations are of the same slope, they are parallel, and the equations form a system of equations with no solution. The correct response is therefore .

There are three variables, but only 2 equations. With fewer equations than variables, it is not possible to solve for the value of each variable separately, but we can eliminate variables to manipulate the equation into the form we want.

Multiply the second equation by  on both sides, resulting in .

Then add the equations together (allowed since we are adding the same thing to both sides, just in a different form):

Simplify:

Resist the urge to keep working, since that's exactly what they've asked for. Done!

Use the addition method to eliminate one variable and therefore solve the system of equations. 

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Now substitute to find x.

Translate the word problem into a system of equations:

Use the first equation to solve for A in terms of O, or vice versa. 

Use substitution to solve for O in the second equation.

Now plug the value for O back into the first equation to solve for A.

Subtract  and  from the both sides to get . 

Divide both sides by , to get

As is clear from the graph, in the interval between  ( included) to , the  is constant at  and then from ( not included) to  ( not included), the  is a decreasing function.