Systems of Linear Equations & Their Solutions: IM Alg1.2.17
[url=https://im.kendallhunt.com/HS/teachers/1/2/17/preparation.html]“Systems of Linear Equations and Their Solutions”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.2.17 Lesson
- IM Alg1.2.17 Lesson: Systems of Linear Equations and Their Solutions
- Alg1.2.17 Practice
- IM Alg.1.2.17 Practice: Systems of Linear Equations and Their Solutions
IM Alg1.2.17 Lesson: Systems of Linear Equations and Their Solutions
Andre is trying to solve this system of equations:
[size=150][math]\begin {cases} x + y = 3\\ 4x = 12 - 4y \end{cases}[/math][/size][br][br][size=150]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."[/size] [br][br]Choose any two numbers that add up to 3. Let the first one be the [math]x[/math]-value and the second one be the [math]y[/math]-value.
The pair of values you chose is a solution to the first equation. Check if it is also a solution to the second equation. Then, pause for a brief discussion with your group.[br]
How many solutions does the system have? Use what you know about equations or about solving systems to show that you are right.
A recreation center is offering special prices on its pool passes and gym memberships for the summer.
[size=150]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. [/size][br][br]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.
Find the price of a pool pass and the price of a gym membership by solving the system algebraically. Explain your reasoning.[br]
Use graphing technology to graph the equations in the system. Make 1-2 observations about your graphs.[br]
Each card contains a system of equations. Sort the systems into three groups based on the number of solutions each system has. Be prepared to explain how you know where each system belongs.
In the cards, for each system with no solution, change a single constant term so that there are infinitely many solutions to the system.
For each system with infinitely many solutions, change a single constant term so that there are no solutions to the system.
Explain why in these situations it is impossible to change a single constant term so that there is exactly one solution to the system.[br]
Here is an equation:
[math]5x-2y=10[/math][br][br]Create a second equation that would make a system of equations with one solution.[br]
Create a second equation that would make a system of equations with no solutions.
Create a second equation that would make a system of equations with infinitely many solutions.
IM Alg.1.2.17 Practice: Systems of Linear Equations and Their Solutions
Here is a system of equations:
[math]\begin{cases} 3x-y=17 \\ x+4y=10 \\ \end{cases}[/math][br][br]Solve the system by graphing the equations (by hand or using technology).[br]
Explain how you could tell, without graphing, that there is only one solution to the system.[br]
Consider this system of linear equations:
[math]\begin{cases} y = \frac45x - 3 \\ y = \frac45x + 1 \end{cases}[/math][br][br]Without graphing, determine how many solutions you would expect this system of equations to have. Explain your reasoning.
Try solving the system of equations algebraically and describe the result that you get. Does it match your prediction?[br]
How many solutions does this system of equations have? Explain how you know.
[math]\displaystyle \begin{cases} 9x-3y=\text-6\\ 5y=15x+10\\ \end{cases}[/math]
[size=150]Select [b]all [/b]systems that have no solutions.[/size]
Solve this system of equations without graphing.
[math]\begin{cases} 2v+6w=\text-36 \\ 5v+2w=1 \end{cases}[/math]
Solve this system of equations without graphing.
[math]\begin{cases} 6t-9u=10 \\ 2t+3u=4 \\ \end{cases}[/math]
[size=150]Select [b]all [/b]the dot plots that appear to contain outliers.[/size]
Here is a system of equations:
[math]\begin{cases} \text-x + 6y= 9 \\ x+ 6y= \text-3 \\ \end{cases}[/math][br][br]Would you rather use subtraction or addition to solve the system? Explain your reasoning.
Here is a system of linear equations:
[math]\begin{cases} 6x-y=18 \\ 4x+2y=26 \\ \end{cases}[/math][br][br]Select [b]all [/b]the steps that would help to eliminate a variable and enable solving.
Consider this system of equations, which has one solution:
[math]\begin {cases} \begin{align} 2x+2y&=180\\0.1x+7y&=\hspace{2mm}78\end{align}\end{cases}[/math][br][br][size=150]Here are some equivalent systems. Each one is a step in solving the original system.[/size][br][br][table][tr][td]Step 1:[/td][td]Step 2:[/td][td]Step 3:[/td][/tr][tr][td][math]\begin {cases} \begin{align} 7x+7y&=630\\0.1x+7y&=\hspace{2mm}78\end{align}\end{cases}[/math][/td][td][math]\begin {cases} \begin{align} 6.9x &=552\\0.1x+7y&=\hspace{2mm}78\end{align}\end{cases}[/math][/td][td][math]\begin {cases} \begin{align} x&=80\\0.1x+7y&=78\end{align}\end{cases}[/math][/td][/tr][/table][br][br][size=150]Look at the original system and the system in Step 1.[/size][br]What was done to the original system to get the system in Step 1?[br]
Explain why the system in Step 1 shares a solution with the original system.[br]
[size=150]Look at the system in Step 1 and the system in Step 2.[br][/size][br]What was done to the system in Step 1 to get the system in Step 2?[br]
Explain why the system in Step 2 shares a solution with that in Step 1.[br]
What is the solution to the original system?[br]