We are given the following system of equations:

We are to find  and . We can solve this through the substitution method.  First, substitute the second equation into the first equation to get

Solve for  by adding 4x to both sides

Add 5 to both sides

Divide by 7

So . Use this value to find  using one of the equations from our given system of equations.  I think I'll use the first equation (can also use the second equation).

So the two linear functions intersect at

Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly"  and "completing assignments" , then we can easily construct a simple system of equations, 

and 

.

We can multiply the first equation by  to yield .

This allows us to cancel the  terms when we add the two equations together. We get , or .

A quick substitution tells us that . So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.

First, combine like terms to get .

Then, subtract 3 and  from both sides to get .

Then, divide both sides by 2 to get a solution of .

To solve this system of equations, the elimination method can be used (the  terms cross out).

Once you eliminate the , you have .

Then isolate for , and you get .

Plug into the first equation to solve for .

Rearrange the second equation:

This is just a multiple of the first equation (by a factor of 3).  Therefore, the two equations are dependent on each are and are going to have an infinite number of solutions.

Solve the first equation for :

Substitute into the second equation:

Multiply the entire equation by 2 to eliminate the fraction:

Using the value of , solve for :

Therefore, the solution is

To solve this system of equations, we must first eliminate one of the variables. We will begin by eliminating the  variables by finding the least common multiple of the  variable's coefficients.  The least common multiple of 3 and 2 is 6, so we will multiply each equation in the system by the corresponding number, like

.

By using the distributive property, we will end up with

Now, add down each column so that you have

Then you solve for  and determine that .

But you're not done yet!  To find , you have to plug your answer for  back into one of the original equations:

Solve, and you will find that .  

Set the two equations equal to one another:

2x - 2 = 3x + 6

Solve for x:

x = -8

Plug this value of x into either equation to solve for y.  We'll use the top equation, but either will work.

y = 2 * (-8) - 2

y = -18

Solve the second equation for y:

x - 2y = 4

-2y = 4 - x

y = -2 + x/2

Plug this into the first equation:

3x + 2(-2 + x/2) = 8

Solve for x:

3x - 4 + x = 8

4x = 12

x = 3

Plug this into the second equation to get a value for y:

3 - 2y = 4

2y = -1

y = -0.5

Start with the equation with the fewest variables, .

Solve for by dividing both sides of the equaion by 6:

Plug this  value into the second equation to solve for :

Subtract 10 from both sides:

Divide by 9:

Plug these and values into the first equation to find :

Combine like terms:

Subtract 2:

Divide by -4:

Therefore the final solution is .