few_shots: 
Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^(2 )+ y^(2) = 3.
problem_activity: Problem 1
A clothing manufacturer has to use at least 1,000 yards of cotton to make room for new inventory to make shirts and pajamas. A shirt requires 1 yard of fabric, and a pair of pajamas requires 3 yards of fabric. It takes 2 hours to make a shirt and 3 hours to make a pair of pajamas, and there are 1600 hours available to make all of the clothing. Let
$$x$$
represent the number of shirts and
$$y$$
represent the number of pajamas.
What is the smallest number of pajamas the manufacturer can make and meet all of these conditions?
Problem 2
Phil and Cath make and sell boomerangs for a school event. The money they raise will go to charity. They plan to make them in two sizes: small and large.
Phil will carve them from wood.
The small boomerang takes 2 hours to carve, and the large one takes 3 hours to carve.
Phil has a total of 24 hours available for carving.
Cath will decorate them. She only has time to decorate 10 boomerangs of either size.
Phil and Cath want to make as much money for charity as they can.
The small boomerang will make $8 for charity.
The large boomerang will make $10 for charity.
How many small and large boomerangs should they make?
How much money will they then make?


aligns: no