standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Activity
Earlier, students saw that adding or subtracting the equations in a system creates a third equation that can help us solve the system. In this activity, students reconnect the idea of the solution to a system to the intersection of the graphs of the equations. They graph each original pair of equations and the equation that results from adding or subtracting them. They then observe that the graph of the third equation intersects the other two graphs at the exact same point—at the intersection of the first two.
At this point, students simply get a graphical confirmation that adding or subtracting equations can help them find the solution to a system. They are not yet expected to be able to articulate why this is the case. That understanding will be developed over a few upcoming lessons.
Launch
Remind students that earlier they added or subtracted pairs of equations to form new equations. Explain that they will now graph each pair of equations in the systems given earlier, as well as the third equation that came from adding or subtracting those equations, and then make some observations about them.
Arrange students into groups of 3 and provide access to graphing technology. Assign one system for each group member to graph. Ask students to discuss their observations after graphing.
If possible, make available graphing technology that allows users to enter linear equations in standard form, such as Desmos (available under Math Tools). Otherwise, give students time to rearrange the equations into a form that can be used with the technology and to check their equivalent equations. If time is limited, provide these equivalent equations:
System A
\(\begin {cases}\begin {align}y &= \text-\frac43x+\frac{10}{3}\\ y&=\hspace{2mm}\frac45x + \frac65 \end{align} \end{cases}\)
System B
\(\begin {cases}\begin {align}y &= \text-2x+4\\ y&=\hspace{3mm}x-11 \end{align} \end{cases}\)
System C
\(\begin {cases}\begin {align}y &= \text-\frac{8}{11}x+\frac{37}{11}\\ y&=\hspace{3.7mm}\text-8x + 7 \end{align} \end{cases}\)
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who are able to find the sum or difference of the equations with ease. If students struggle, direct students to use parentheses around the entirety of the equation being added or subtracted to enhance clarity and reduce errors.
Supports accessibility for: Memory; Organization
Student Facing
Here are three systems of equations you saw earlier.
System A
\(\begin {cases}\begin {align}4x + 3y &= 10\\ \text-4x + 5y &= \hspace{2mm}6 \end{align} \end{cases}\)
System B
\(\begin {cases} \begin {align}2x + y &= 4\\ x - y &= 11 \end {align} \end {cases} \)
System C
\(\begin {cases} \begin{align} 8x + 11y &= 37\\ 8x + \hspace{4mm} y &= \hspace{2mm} 7 \end{align} \end{cases}\)
For each system:
Use graphing technology to graph the original two equations in the system. Then, identify the coordinates of the solution.
Find the sum or difference of the two original equations that would enable the system to be solved.
Graph the third equation on the same coordinate plane. Make an observation about the graph.
Student Response
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Student Facing
Are you ready for more?
Mai wonders what would happen if we multiply equations. That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.
In system B write out an equation that you would get if you multiply the two equations in this manner.
Does your original solution still work in this new equation?
Use graphing technology to graph this new equation on the same coordinate plane. Why is this approach not particularly helpful?
Student Response
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Anticipated Misconceptions
When solving system B, some students may not notice that the
\(y\)
-variable in one equation has a positive coefficient and the other has a negative coefficient, and consequently decide to subtract the second equation from the first, rather than to add the two equations. They may struggle to figure out why the solution pair they find doesn't match what is on the graph. Suggest that they express the second equation in terms of addition,
\(x+ (\text-y) = 11\)
, and try eliminating one variable again.
Activity Synthesis
Display the graphs that students generated for all to see and ask students to share their observations. Highlight that the graph of the new equation intersects the graphs of the equations in the original system at the same point.
Time permitting, ask students to subtract the equations they previously added (or add the equations they previously subtracted) and then to graph the resulting equation on the same coordinate plane. Ask them to comment on the graphs. Students are likely to see that the graphs of the new equations are no longer horizontal or vertical lines, but they still intersect at the same point as the original graphs.
Invite students to make some conjectures as to why the graph of the new equation intersects the other two graphs at the same point. Without confirming or correcting their conjectures, tell students that they will investigate this question in the coming activities.


aligns: no