We can solve this problem by elimination. 

Subtract the second equation from the first to eliminate the x values, like so:

This yields

Now, substitute  into either equation and solve for x. In either equation, we find that . 

Therefore, the solution to this system of equations is .

To solve this problem, we can set up a system of equations. 

Let  represent the cost of donuts, and let  represent the cost of a cup of coffee. 

We can write Giovanni and Jalynn's purchases as equations like this:

To solve this system of equations, we can use elimination by subtracting the second equation from the first, like so:

Our  terms drop out, leaving us with

Therefore, each cup of coffee costs . 

Because the coefficients of the  values are the same, we can solve this problem by elimination. Subtract the second equation from the first:

This knocks out our  value and gives us , or .

Then, substitute this value into either of the equations to solve for . Let's use the second equation:

Therefore, the solution is 

Since none of our coefficients are the same, we cannot solve by elimination. Therefore, we'll use substitution. Let's solve for one term in one equation. 

Now, we can substitute this expression into the second equation.

Now, we can substitute this value back into either of the original equations to find our solution. Let's use the simplest equation:

With these values, we see that our solution is the point . 

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!