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 problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Activity
In this activity, students find intercepts and slopes for pairs of equations. They should notice that the equations in each pair are multiples of one another and therefore are equivalent equations. Students look for and make use of structure (MP7) when they notice the relationships between the pairs of equations.
Launch
Ask students what they remember about
\(x\)
- and
\(y\)
-intercepts. One useful thing to remember is that the
\(x\)
-intercept of a graph occurs where the
\(y\)
-coordinate is zero. (Demonstrate using a graph, if needed.) So, to find the
\(x\)
-intercept of a graph representing an equation, you can replace
\(y\)
with 0 and solve the equation for
\(x\)
. (And vice versa to find the
\(y\)
-intercept.)
Ask students how they can use two points like
\((1,0)\)
and
\((0,8)\)
to find the slope of a line. (Using a formula like
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
, or a less-formal version like “subtract the y’s, subtract the x’s, then divide.
”
For this example the slope is -8 since
\(\frac{8 - 0}{0-1} = \text{-}8\)
.)
Student Facing
Here are some equations in pairs. For each equation:
Find the
\(x\)
-intercept and
\(y\)
-intercept of a graph of the equation.
Find the slope of a graph of the equation.
\(x + y = 6\)
and
\(2x + 2y = 12\)
\(3y - 15x = \text{-}33\)
and
\(y - 5x = \text{-}11\)
\(5x + 20y = 100\)
and
\(4x + 16y = 80\)
\(3x - 2y = 10\)
and
\(4y - 6x = \text{-}20\)
What do you notice about the pairs of equations?
Choose one pair of equations and rewrite them into slope-intercept form (
\(y = mx + b\)
). What do you notice about the equations in this form?
Student Response
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Activity Synthesis
The purpose of the discussion is to show that equations that are multiples of one another will have graphs with the same intercepts and slope. Select students to share their solutions and methods for finding the intercepts and slope as well as things they notice. While students may notice many things, ensure it is mentioned that in each pair of equations, one is a multiple of the other.
Some questions for discussion:
"How would the graphs of each pair of equations look?" (Each graph would look like a single line since the two equations are equivalent. Time permitting, demonstrate this using graphing technology.)
"Select a pair of equations. Find another equation whose graph would have the same slope and intercepts as the pair already given." (Another multiple of either equation will work.)


aligns: yes