ACTIVITY: Types of Solutions – Consistent or Inconsistent
PROBLEM 1: Saving Up
Marcus and Phillip are in the Robotics Club. They are both saving money to buy materials to build a new robot.
They plan to save the same amount of money each week.
Marcus decides to open a new bank account. He deposits $25 that he won in a robotics competition. He also
plans on depositing $10 each week that he earns from tutoring. Phillip decides he wants to keep his money in a
sock drawer. He already has $40 saved from mowing lawns over the summer. He plans to save $10 a week
from his allowance.
1. Define your variables.
2. Write equations to represent the information regarding Marcus and Philip saving money for new robotics
materials.
3. Predict when Marcus and Phillips will have the same amount of money saved. Use your equations to help
you determine your prediction. (I do not what to see any math. I just want you think based on the equations!)
Now, we will work to provide your prediction by solving and graphing the system of equations you wrote for
Marcus and Phillip.
4.
Analyze each equation.
a. Describe what the slope of each line represents in this situation.
b. How do the slopes compare (greater than, less than, or equal to)? Describe what this means in terms of
this problem.
c.
Describe what the y-intercept of each line represents in this problem.
d.
How do the y-intercepts compare (greater than, less than, or equal to)? Describe what this means in
terms of this problem.
Taken From Carnegie Learning Integrated Mathematics I
ACTIVITY: Types of Solutions – Consistent or Inconsistent
5.
Determine the solution to the system of linear equations algebraically and graphically.
a.
Use substitution/equal values to determine the intersection point. (Think: What is the point of intersection
called?)
b.
Does your solution make sense? Describe what this means in terms of the problem.
c.
Predict what the graph of this system will look like. Explain your reasoning.
Graph 1
d. Graph both equations on the coordinate plane. (LABEL!!)
6.
Analyze the graph you created.
a.
Describe the relationship between the graphs.
b. Does this linear system have a solution?
Explain your reasoning.
7. How is what you see in your graph connected to
what you got when you solved the problem using
substitution?
8. Was your prediction in Question 3 correct? Explain how you proved or disproved your prediction.
Taken From Carnegie Learning Integrated Mathematics I
ACTIVITY: Types of Solutions – Consistent or Inconsistent
PROBLEM 2: Tonya’s Savings Plan
Tonya is also in the Robotics Club and has learned about Marcus’s and Phillip’s (from the first problem) savings
plans. She wants to be able to buy her new materials before Phillip, so she opens her own bank account. She is
able to deposit $40 in her account that she has saved from her job as a waitress. Each week also deposits $4
from her tips.
Graph 2
9. Define your variables.
10. Write an equation that represents the information about
Tonya saving money every week.
11. Write the equations for Tonya and Phillip
as a systems of equations.
12. Create a table for Tonya’s equation. You will need it to create the graph.
13. Using your table in #12, graph Tonya’s equation. You can use the slope and y-intercept to graph Phillips’s
equation.
14. Do the graphs intersect? If so, describe the meaning in terms of this problem.
Taken From Carnegie Learning Integrated Mathematics I
ACTIVITY: Types of Solutions – Consistent or Inconsistent
15. Phillip and Tonya went on a shopping spree this weekend and spent all their savings except for $40 each. (so
both Tonya and Phillip only have $40). Phillip is still saving $10 a week from his allowance. Tonya now
deposits her tips twice a week. Each Tuesday she deposits $4 and each Saturdays she deposits $6. Phillips
claims he is still saving more each week than Tonya.
a.
Do you think Phillips’s claim is true? Explain your reasoning.
b. Rewrite the equation for Tonya based on her “new” savings plan. Also, rewrite Phillip’s equation here as
well.
Graph 3
c.
What do you notice about the two equations?
d.
Graph the new equations. LABEL!
13. Analyze the graph.
a. Describe the relationship between the graphs.
What does this mean in terms of this problem?
b. Verify algebraically what you have found graphically.
c. Does this solution prove the relationship? Explain your reasoning.
d.
Was Phillip’s claim that he is still saving more than Tonya a true statement? Explain why or why not.
Taken From Carnegie Learning Integrated Mathematics I
ACTIVITY: Types of Solutions – Consistent or Inconsistent
A system of equations may have one unique solution (a point of intersection), infinitely many solutions (same
line) or no solutions (parallel). Systems that have one or many solutions are called consistent systems.
Systems with no solution are called inconsistent systems.
CONSISTENT SYSTEMS
INDEPENDENT
Slope
Description of
y-intercepts
Description of
Graph
Number of
Solutions
Algebraically
Taken From Carnegie Learning Integrated Mathematics I
DEPENDENT
INCONSISTENT
SYSTEMS
ACTIVITY: Types of Solutions – Consistent or Inconsistent
SUMMARY:
When you solve a ___________ of ___________, you can get ___________ different outcomes. One
outcomes is no ___________. This means that the lines ____________________ so the lines are ___________. No point of ___________ means there is ______________. You can tell this will happen
because the ___________ of the lines will be the ___________ but the ___________ will be ___________.
If you solved the system ___________, you would get two___________ numbers such as ___________.
Since this is ___________ true, there are no ___________. These systems are i___________.
Another outcome is to have ___________ number of solutions. This means that you have___________
of the ___________ equation. When graphed, you will graph the ___________ line twice. This means that
each ___________ on one___________ is on the other___________. Since each line has an ___________
number of ___________, then there are___________ number of ___________ solutions. You can tell this will
happen because the ___________ and ___________ will both be the ___________. If you solve the system
___________, you would get two ___________ numbers such as ___________. Since this is ___________
true, there are an ___________ number of___________. These systems are ___________ and ___________.
Another outcome is to have only ___________ solution. This means you have ___________ different
___________. When graphed, you will have ___________ lines which ___________ i. This means that the
___________ share only ___________ point or ___________. You can tell this will happen because the
___________ will have ___________ slopes. The ___________ may be ___________ but could be the
same if the lines ___________ at the ___________ it will be the ___________ for both lines. If you solve the
system___________, you would get x ___________ a number such as ___________. You would then have
to___________ the value of ___________ into one of the ___________ to find the value of___________.
These systems are___________ and ___________.
Taken From Carnegie Learning Integrated Mathematics I