Student handout: Systems of Equations Compare and Contrast

Systems of Equations: Compare and Contrast

Small Group Directions

1. Solve your assigned system of equations using any algebraic method. Show you work and be ready to explain how you solved it.
2. Also graph the system of equations and show how the solution appears on your graph. You may use graphing technology such as Desmos.

Group Roles

Facilitator: Read the directions out loud and check whether everyone understands each other. “How should we start?” “How do you know?”

Team Captain: Help your team members step up and step back. “How do you know?” “What do you think?”

Recorder/Reporter: Be prepared to share out in the whole class discussion. “How should I explain...?”

Problems

1. $y=-3x\\4x+y=2$
2. $y=7x-5\\2x+y=13$
3. $x=-5y-4\\x-4y=23$
4. $x+y=10\\y=x-4$
5. $y=5-x\\4x+2y=10$
6. $3x+5y=23\\y=x+3$
7. $y=-x-2\\2x+3y=-9$
8. $y=2x-3\\-2x+y=1$
9. $x=\frac{1}{2}y+\frac{1}{2}\\2x+y=-1$
10. $a=2b+4\\b-2a=16$
11. $y=3-2x\\4x+2y=6$
12. $y=x+1\\x-y=1$

Whole Class Directions

1. Each group will share out how you solved your system of equations.
2. Listen to each group and think about similarities and differences.
3. Ask questions about anything you do not understand or you disagree with.
4. You do not need to write anything during the whole class discussion, but you will have an exit ticket to see what you learned from the discussion.

Exit Ticket: Systems of Equations Compare and Contrast

Sheila missed class today. She tried to solve Problem 8 on her own, but she thinks she made a mistake because -3 does not equal 1. \begin{align} &y=2x-3\\ &-2x+y=1 \end{align} \begin{align} &-2x+(2x-3)=1\\ &-2x+2x-3=1\\ &0-3=1\\ &-3=1 \end{align}

Explain to Sheila what happened, using as much detail as possible to help her understand this type of problem.

• assignment Differentials of One Variable

assignment Homework

Differentials of One Variable
Static Fields 2022 (5 years) Find the total differential of the following functions:
1. $y=3x^2 + 4\cos 2x$
2. $y=3x^2\cos kx$ (where $k$ is a constant)
3. $y=\frac{\cos 7x}{x^2}$
4. $y=\cos(3x^2-2)$
• assignment Center of Mass for Two Uncoupled Particles

assignment Homework

Center of Mass for Two Uncoupled Particles
Central Forces 2023 (3 years)

Consider two particles of equal mass $m$. The forces on the particles are $\vec F_1=0$ and $\vec F_2=F_0\hat{x}$. If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

• assignment Wavefunctions

assignment Homework

Wavefunctions
Quantum Fundamentals 2022 (2 years)

Consider the following wave functions (over all space - not the infinite square well!):

$\psi_a(x) = A e^{-x^2/3}$

$\psi_b(x) = B \frac{1}{x^2+2}$

$\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)$ (“sech” is the hyperbolic secant function.)

In each case:

1. normalize the wave function,
2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
3. find the probability that the particle is measured to be in the range $0<x<1$.

• assignment Paramagnet (multiple solutions)

assignment Homework

Paramagnet (multiple solutions)
Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization $M$, and the entropy $S$ of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: $\chi_B=\left(\frac{\partial M}{\partial B}\right)_T$

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

$\left(\frac{\partial M}{\partial B}\right)_S$

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?

• group Paramagnet (multiple solutions)

group Small Group Activity

30 min.

Paramagnet (multiple solutions)
• Students evaluate two given partial derivatives from a system of equations.
• Students learn/review generalized Leibniz notation.
• Students may find it helpful to use a chain rule diagram.
• assignment Zapping With d 1

assignment Homework

Zapping With d 1
Energy and Entropy 2021 (2 years)

Find the differential of each of the following expressions; zap each of the following with $d$:

1. $f=3x-5z^2+2xy$

2. $g=\frac{c^{1/2}b}{a^2}$

3. $h=\sin^2(\omega t)$

4. $j=a^x$

5. $k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)$

• group Applying the equipartition theorem

group Small Group Activity

30 min.

Applying the equipartition theorem
Contemporary Challenges 2022 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• assignment Divergence Practice including Curvilinear Coordinates

assignment Homework

Divergence Practice including Curvilinear Coordinates

Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

1. $$\hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\hat{K} = s^2\,\hat{s}$$
7. $$\hat{L} = r^3\,\hat{\phi}$$

• assignment Curl Practice including Curvilinear Coordinates

assignment Homework

Curl Practice including Curvilinear Coordinates

Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

1. $$\vec{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$$
2. $$\vec{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$$
3. $$\vec{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$$
4. $$\vec{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$$
5. $$\vec{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$$
6. $$\vec{K} = s^2\,\hat{s}$$
7. $$\vec{L} = r^3\,\hat{\phi}$$

• assignment Theta Parameters

assignment Homework

Theta Parameters
Static Fields 2022 (5 years)

The function $\theta(x)$ (the Heaviside or unit step function) is a defined as: $$\theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases}$$ This function is discontinuous at $x=0$ and is generally taken to have a value of $\theta(0)=1/2$.

Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}

Author Information
Alyssa Sayavedra
Learning Outcomes