Unit 4 Linear Equations And Linear Systems — Unit Plan

Title

Assessment

Lesson 1

Writing Equivalent Equations

Explain the Reasoning

Label all 4 arrows to describe what happens in each move.
Are the equations equivalent? Explain your reasoning.

Show Solution

Subtract 3 (or add -3) and divide by 2 (or multiply by $\frac{1}{2}$ )
Yes. Sample reasoning: As long as the same operations are done correctly to each side, the equations remain equivalent.

Lesson 2

Keeping the Equation Balanced

Changing Blocks

Here is a hanger that is in balance. We don’t know how much any of its shapes weigh.

<p>A balanced hanger. Left side, 2 circles, 4 squares. Right side, 2 squares, 2 triangles, 2 circles, 2 triangles.</p><br>

How could you remove shapes from the hanger and keep it in balance? Describe in words or draw a new diagram.
How could you add shapes to the hanger and keep it in balance? Describe in words or draw a new diagram.

Show Solution

Sample response:

I can remove 2 circles from each side.
I can add 1 triangle to each side.

Lesson 3

Balanced Moves

More Matching Moves

Match these pairs of equations with the description of what is done in each step.

Step 1:

$\begin{align} 12x-6&=10\\ 6x-3&=5 \end{align}$

A:

Add 3 to each side

Step 2:

$\begin{align} 6x-3&=5\\ 6x&=8 \end{align}$

B:

Multiply each side by $\frac16$

Step 3:

$\begin{align} 6x&=8\\ x&=\frac43 \end{align}$

C:

Divide each side by 2
You are given the equation $3(x-2) = 8$ . Is your first step to distribute or divide? Explain your reasoning.

Show Solution

Step 1: C, Step 2: A, Step 3: B
Sample responses:
- I would distribute the 3. That way I do not need to deal with fractions like $\frac{8}{3}$ until the end.
- I would divide each side by 3. Then there are fewer terms to manage while solving.

Lesson 4

More Balanced Moves

Mis-Steps

Examine Lin’s solution to $8(x-3) + 7 = 2x(4-17)$ .

Lin’s solution:

For each step, determine if the 2 equations are equivalent. If they are not, describe the error.
What is the correct solution to the original equation?

Show Solution

Sample response:

$x = \frac{1}{2}$ or equivalent

Lesson 5

Solving Any Linear Equation

Check It

Noah tries to solve the equation $\frac{1}{2}(7x-6)=6x-10$ .

Check Noah’s work. If it is not correct, describe what is wrong and show the correct work.

$\begin{align} \frac{1}{2}(7x - 6) &=6x - 10 \\[2ex] 7x - 6 &=12x - 10 \\[2ex] 7x &= 12x - 4 \\[2ex] \text{-}5x &= \text{-}4 \\[2ex] x &= \frac{4}{5} \end{align}$

Show Solution

Sample response: Going from line 1 to line 2, Noah tried to multiply each side of the equation by 2, but did not multiply the 10. When you double each side of an equation, each term needs to be multiplied by 2.

$\begin{align} \frac{1}{2}(7x - 6) &=6x - 10 \\ 7x - 6 &=12x - 20 \\ 7x &= 12x - 14 \\ \text{-}5x &= \text{-}14 \\ x &= \frac{14}{5} \end{align}$

Lesson 6

Strategic Solving

Think Before You Step

Without solving, identify whether this equation has a solution that is positive, negative, or zero. Explain your reasoning.

$3x-5=\text-3$
Solve the equation.

$x-5(x-1)=x-(2x-3)$

Show Solution

Positive. Sample reasoning: If $3x-5=\text-3$ , then the $x$ must be positive. If $x$ is negative, then subtracting 5 from $3x$ would result in a number less than $\text-3$ . For similar reasons, $x$ cannot be zero.
$x=\frac23$ (or equivalent)

Section A Check

Section A Checkpoint

Problem 1

Label the arrows to describe the moves that create equivalent equations.
Are these 2 equations equivalent? Explain your reasoning.

$\begin{align} 4x + 2 &= 20x\\ x + 2 &= 5x \end{align}$

Show Solution

Add 3, distributive property
No. Sample reasonings:
- Each term on the left should be divided by 4, but the 2 was not divided.
- The solutions are not the same. For the second equation, the solution is $x = \frac{1}{2}$ , but that does not solve the first equation because $4 \boldcdot \frac{1}{2} + 2 \neq 20\boldcdot \frac{1}{2}$ .

Problem 2

Solve the equation. Show your reasoning by describing any moves that you make to write equivalent equations.

$4(3 - x) = 3x-2$

Show Solution

Sample response:

$12 - 4x = 3x - 2$ , I applied the distributive property

$12 = 7x - 2$ , I added $4x$ to each side

$14 = 7x$ , I added 2 to each side

$2 = x$ , I divided each side by 7

Lesson 7

All, Some, or No Solutions

Choose Your Own Solution

$\displaystyle 3x + 8 = 3x + \underline{\hspace{.5in}}$

What value could you write in after " $3x$ + " that would make the equation true for:

no values of $x$ ?
all values of $x$ ?
just one value of $x$ ?

Show Solution

Any value other than 8.
8
Any variable term. like $x$ or $2x$ , in order to create an equation with one solution.

Lesson 8

How Many Solutions?

How Does She Know?

Elena begins to solve this equation:

$\begin{align} \dfrac{12x+6(4x+3)}3 &\,=\,2(6x+4)-2 \\[2ex]12x+6(4x+3) &\,=\,3(2(6x+4)-2)\\[2ex] 12x+6(4x+3) &\,=\,6(6x+4)-6 \\[2ex] 12x+24x+18 &\,=\,36x+24-6 \end{align}$

When she gets to the last line she stops and says the equation is true for all values of $x$ . How can Elena tell?

Show Solution

Sample response: Elena can see that there are the same number of $x$ 's and the same constant terms on each side of the equation.

Lesson 9

When Are They the Same?

Printers and Ink

To own and operate a home printer, it costs $100 for the printer and an additional $0.05 per page for ink. To print out pages at an office store, it costs $0.25 per page. Let $p$ represent number of pages.

What does the equation $100+0.05p=0.25p$ represent?
The solution to that equation is $p=500$ . What does the solution mean?

Show Solution

The equation represents when the cost for owning and operating a home printer is equal to the cost for printing at an office store.
The solution of $p=500$ means that the costs are equal for printing 500 pages.

Section B Check

Section B Checkpoint

Problem 1

$3x + 7 = 5x + 7$

How many solutions does the equation have? Explain how you know without solving.
Change 1 number in the equation $2x + 4 = 2x + 6$ so that it has infinitely many solutions.

Show Solution

1 solution. Sample reasoning: The coefficients of $x$ on each side of the equation are not equal.
Sample responses:
- $2x + 4 = 2x+4$
- $2x + 6 = 2x+6$

Problem 2

Two friends go out for a run.

Friend A runs at a steady pace of 160 meters per minute so that their distance from the starting line is represented by $160t$ .
Friend B gets started later and begins running a little further along the route so that their distance from the starting line is represented by $180(t-3)+100$ .

Solve the equation $160t = 180(t-3)+100$ . Show your reasoning.
What does the solution mean in this situation?

Show Solution

$t = 22$ . Sample reasoning: $160t=180t - 540 + 100$ by distributive property. $160t = 180t - 440$ by combining like terms. $\text{-}20t = \text{-}440$ by subtracting $180t$ from each side. $t = 22$ by dividing each side by -20.
Sample response: 22 minutes after Friend A started running the friends are the same distance from the starting line.

Lesson 11

On Both of the Lines

Saving Cash

Andre and Noah start tracking their savings at the same time.

Andre starts with $15 and deposits $5 per week.

Noah starts with $2.50 and deposits $7.50 per week. The graph of Noah's savings is given, and his equation is $y=7.5x+2.5$ , where $x$ represents the number of weeks and $y$ represents his savings.

Write the equation for Andre's savings, and graph it alongside Noah's. What does the intersection point mean in this situation?

<p>Graph of a line in the x y plane.</p><br>

Show Solution

Sample response:

The intersection at $(5,40)$ means that after 5 weeks, Noah and Andre each have $40.

Lesson 12

Systems of Equations

Finishing Their Water Again

Lin’s glass has 12 ounces of water and she drinks it at a rate of $\frac{1}{3}$ ounce per second.

Diego’s glass has 20 ounces and he drinks it at a rate of $\frac{2}{3}$ ounce per second.

Graph this situation on the axes provided.
What does the graph tell you about the situation and how many solutions there are?

Show Solution

Sample response: There is one solution at $(24,4)$ meaning that after 24 seconds both of them have 4 ounces of water left.

Lesson 14

Solving More Systems

Solve It

Solve this system of equations:

$\begin{cases} y=2x \\[2ex] x = \text-y+6 \end{cases}$

Show Solution

$(2,4)$ . Sample Reasoning: Use the substitution method to rewrite the system as the one variable equation $x = \text{-} (2x)+6$ , then solve.

Lesson 15

Writing Systems of Equations

Solve This

Solve.

$\begin{cases} y= \dfrac34x \\[2ex] \dfrac52x+2y = 5 \end{cases}$

Show Solution

$x = \frac{5}{4}, y = \frac{15}{16}$

Section C Check

Section C Checkpoint

Problem 1

$\begin{cases} y = 3x + 5 \\ y = 3(x + 1) \end{cases}$

How many solutions does this system have? Explain your reasoning without solving the system.
Based on the number of solutions, describe the graph of this system.

Show Solution

No solutions. Sample reasoning: The second equation is equivalent to $y = 3x + 3$ . This shows that the 2 equations have the same slope and different $y$ -intercepts, so there is no solution.
The graphs of the lines are parallel.

Problem 2

In a card game, each round you earn either 3 points or 5 points depending on the cards you play. After 5 rounds you have 19 points.

Use $x$ for the number of 3 point rounds and $y$ for the number of 5 point rounds. Write a system of 2 equations that describes this situation.
Another system is solved by the point $(7,10)$ . Explain how you can check that this solution is correct.

Show Solution

$\begin{cases} 3x + 5y &= 19\\ x + y &= 5 \end{cases}$ (or equivalent)
Sample response: The values make both equations true. Substitute 7 for $x$ and 10 for $y$ in the original equations and check that each side of the equations are equal to the other side.

Lesson 16

Solving Problems with Systems of Equations

No cool-down

Unit 4 Assessment

End-of-Unit Assessment

Problem 1

Select all of the equations that are equivalent to

2x + 6 = x - 4

x + 6 = \text{-}4

2x = x + 2

2x + 8 = x - 2

2(x+3) = x - 4

2x + 3 = x - 2

Show Solution

A, C, D

Problem 2

Select all the systems of equations that have exactly 1 solution.

$\begin{cases} y = 3x + 1 \\ y = \text-3x - 7 \end{cases}$

$\begin{cases} y = 3x + 1 \\ y = x + 1 \end{cases}$

$\begin{cases} y =3x + 1 \\ y = 3x + 7 \end{cases}$

$\begin{cases} x+y = 10 \\ 2x + 2y = 20 \end{cases}$

$\begin{cases} x + y = 10 \\ x + y = 12 \end{cases}$

Show Solution

A, B

Problem 3

Which system of equations has a solution of $(3,\text{-}4)$ ?

\begin{cases} y = 2x -10 \\ y = \text{-}x +1 \end{cases}

\begin{cases} x + y = \text{-}1 \\ y = \text{-}4x + 8 \end{cases}

\begin{cases} y = 4 - 2x \\ y + 3x = 5 \end{cases}

\begin{cases} y - 5x = \text{-}17 \\ x - y = \text{-}7 \end{cases}

Show Solution

\begin{cases} x + y = \text{-}1 \\ y = \text{-}4x + 8 \end{cases}

Problem 4

Solve this equation. Explain or show your reasoning.

$\displaystyle \frac 1 2 x - 7 = \frac 1 3 \left(x - 12\right)$

Show Solution

$x = 18$ . Sample reasoning: Use the distributive property to rewrite the equation as $\frac 1 2 x - 7 = \frac 1 3 x - 4$ . Then, subtract $\frac 1 3 x$ from each side: $\frac 1 6 x - 7 = \text{-}4$ . Add 7 to each side: $\frac 1 6 x = 3$ . Then $x = 3 \div \frac 1 6 = 3 \boldcdot 6 = 18$ .

Minimal Tier 1 response:

Work is complete and correct.
Sample:
$\frac 1 2 x - 7 = \frac 1 3 x - 4$
$\frac 1 6 x - 7 = \text{-}4$
$\frac 1 6 x = 3$

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Algebra mistakes not directly related to the work of this unit: incorrectly subtracting or dividing fractions; incorrectly adding or subtracting integers.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Solution given with no work shown; algebra mistakes that are pertinent to the work of this unit: dividing both sides of the equation by $\frac13$ initially, but dividing only $\frac12x$ or -7 by $\frac13$ ; incorrect use of the distributive property; failure to use inverse operations.

Problem 5

Solve this system of equations.

$\begin{cases} 3x + 4y = 36\\ y=\text-\frac{1}{2} x + 8 \end{cases}$

Show Solution

$x = 4$ , $y = 6$

Problem 6

Andre and Elena are each saving money. Andre starts with $100 in his savings account and adds $5 per week.

The amount of money Elena has saved is shown by the line in this graph

Write an equation representing Andre’s savings after $x$ weeks.
After how many weeks will Andre and Elena have the same amount of money in their savings accounts? Explain or show your reasoning.

Show Solution

$y = 5x+100$ (or equivalent)
6 weeks. Sample reasonings:
- Graphing Andre’s savings on the same graph as Elena, I see that the lines intersect at $(6,130)$ , so they each have $130 after 6 weeks.
- An equation for Elena’s savings is $y = 20x+10$ . Solve $5x+100 = 20x+10$ to get $x = 6$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$y = 5x + 100$
6 weeks. Correct graph of Andre’s equation and intersection point marked (or correctly set up and solved one-variable equation).

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Andre’s equation is written as an expression; algebra or graphing mistakes in Part B work contains a correct solution to the equation in Part B, but the explanation does not answer the number of weeks.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: No sensible approach to writing Andre's equation; no reasonable equations, tables, or graphs to represent amount of money saved after $x$ weeks; no reasonable method for finding the number of weeks after which Andre and Elena will have the same amount of money.

Problem 7

At a game night, people can choose to play chess, a 2-player game, or to play hearts, a 4-player card game.

60 people are playing the games with $x$ representing the number of chess games being played and $y$ representing the number of hearts games.

Complete this table showing some possible combinations of the number of each type of game being played.

chess games ( $x$ ) hearts games ( $y$ )

30 0

8

10

15

2
There are 3 more games of hearts being played than games of chess being played. How many of each game are being played? Explain or show your reasoning.

Show Solution

chess games ( $x$ ) hearts games ( $y$ )

30 0

14 8

10 10

0 15

2 14
8 chess games and 11 hearts games. Sample reasoning: The solution solves the system of equations $2x + 4y = 60$ and $y = x + 3$ . Solve by substitution: $2x + 4(x+3) = 60$ . Then $x = 8$ , and $y = 11$ because $y = x + 3$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Solutions that simply involve filling out more rows of the table are acceptable.
Sample:

See table.
Solve the system $2x + 4y = 60$ and $y = x + 3$ .
$2x + 4(x+3) = 60$
$2x+4x+12=60$
$6x=48$
$x=8$
$y=11$ .
8 chess games, 11 hearts.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: One row of the table is incorrect; system of equations is present but work to solve those equations contains algebra errors; equation to represent “3 more games of hearts than chess” actually represents “3 more games of chess than hearts.”

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: One of the equations in Part B is incorrect because of an error filling out the table in Part A.
Sample errors: Several rows of the table are incorrect; a system of equations is present and their solution is correct, but the equations do not come close to representing the situation; a correct system of equations is present but the work is incorrect and does not involve the substitution method.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrect or missing answer to Part B with no system of equations written; misinterpretation of the 4-player game/2-player game constraint means that the table is filled out completely incorrectly.