Learning Targets
I can tell how many solutions a system has by graphing the equations or by analyzing the parts of the equations and considering how they affect the features of the graphs.
I know the possibilities for the number of solutions a system of equations could have.
Andre is trying to solve this system of equations:
Looking at the first equation, he thought, "The solution to the system is a pair of numbers that add up to 3. I wonder which two numbers they are."
1. Choose any two numbers that add up to 3. Let the first one be the
2. The pair of values you chose is a solution to the first equation. Check if it is also a solution to the second equation.
3. How many solutions does the system have?
Explain your answer for question 3.
A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72.
1. Write a system of equations that represents the relationships between pool passes, gym memberships, and the costs. Be sure to state what each variable represents.
A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72.
2. Find the price of a pool pass and the price of a gym membership by solving the system algebraically. Explain or show your reasoning.
A recreation center is offering special prices on its pool passes and gym memberships for the summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. She paid $72.
3. Use graphing technology to graph the equations in the system.
Make 1-2 observations about your graphs.
How can we tell from the graphs that there are no solutions?
How can we tell for sure that the lines are parallel and never intersect?
Why do parallel lines mean no solutions?
What does ‘no solutions’ mean in this situation, in terms of price of pool passes and gym memberships?
Sort the systems into three groups based on the number of solutions each system has.
One Solution
Infinitely Many Solutions
No Solutions
How can we tell that the systems have no solution?
What features might give us a clue that the systems have many solutions?
Here is an equation:
Create a second equation that would make a system of equations with one solution
Here is an equation:
Create a second equation that would make a system of equations with no solutions
Here is an equation:
Create a second equation that would make a system of equations with infinitely many solutions
Mai is given these two systems of linear equations to solve:
System 1:
System 2:
She analyzed them for a moment, and then—without graphing the equations—said, "I got it! One of the systems has no solution and the other has infinitely many solutions!" Mai is right!
Which system has no solution and which one has many solutions? Explain or show how you know (without graphing the equations).