If we use elimination to solve this system of equations, we can add the two equations together. This results in 0=0. 

When elimination results in 0=0, that means that the two equations represent the same line. Therefore, there are an infinite number of solutions. 

We can solve this system of equations by elimination since the 2 given y-values have the same coefficient. Let's subtract the second equation from the first.

This gives us  or .

Substitute this x-value into either equation and solve for y. Let's use the first equation like so:

The solution is .

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 :