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: Distinguish between situations that can be modeled with linear functions and with exponential functions.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
| $x$ | $f(x)=128\cdot \left({\frac {1}{2}}\right)^x$ |               Successive quotients                |
|-----|-----------------------------------------------|---------------------------------------------------|
|  0  |                      128                      |                       ----                        |
|  1  |                      64                       | $\frac{f(1)}{f(0)}=\frac {64}{128} = \frac{1}{2}$ |
|  2  |                                               |               $\frac{f(2)}{f(1)}=$                |
|  3  |                                               |                                                   |
|  4  |                                               |                                                   |
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
| $x$ | $f(x)=128\cdot \left({\frac {1}{2}}\right)^x$ |                Successive quotients                 |
|-----|-----------------------------------------------|-----------------------------------------------------|
|  0  |                      128                      |                         ---                         |
|  2  |                      32                       | $\frac{f(2)}{f(0)} = \frac {32}{128} = \frac{1}{4}$ |
|  4  |                                               |                $\frac{f(4)}{f(2)}=$                 |
|  6  |                                               |                                                     |
|  8  |                                               |                                                     |
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


aligns: yes