Unit 4: Practice Problem Sets

Tyler reads $\frac{2}{15}$ of a book on Monday, $\frac13$ of it on Tuesday, $\frac29$ of it on Wednesday, and $\frac34$ of the remainder on Thursday. If he still has 14 pages left to read on Friday, how many pages are there in the book?

Clare asks Andre to play the following number puzzle:

• Pick a number

• Add 2

• Multiply by 3

• Subtract 7

• Add your original number

Andre’s final result is 27. Which number did he start with?

In a basketball game, Elena scores twice as many points as Tyler. Tyler scores four points fewer than Noah, and Noah scores three times as many points as Mai. If Mai scores 5 points, how many points did Elena score? Explain your reasoning.

A store is designing the space for rows of nested shopping carts. Each row has a starting cart that is 4 feet long, followed by the nested carts (so 0 nested carts means there's just the starting cart). The store measured a row of 13 nested carts to be 23.5 feet long, and a row of 18 nested carts to be 31 feet long.

1. Create a graph of the situation.
   
2. How much does each nested cart add to the length of the row? Explain your reasoning.
    
3. If the store design allows for 43 feet for each row, how many total carts fit in a row?
    

What is the weight of a square if a triangle weighs 4 grams?

Andre came up with the following puzzle. “I am three years younger than my brother, and I am 2 years older than my sister. My mom's age is one less than three times my brother's age. When you add all our ages, you get 87. What are our ages?”

1. Try to solve the puzzle.

2. Jada writes this equation for the sum of the ages: $(x)+(x+3)+(x-2) + 3(x+3) - 1=87$. Explain the meaning of the variable and each term of the equation.

3. Write the equation with fewer terms.

4. Solve the puzzle if you haven’t already.

These two lines are parallel. Write an equation for each.

In this hanger, the weight of the triangle is $x$ and the weight of the square is $y$.

Andre and Diego were each trying to solve $2x+6=3x-8$. Describe the first step they each make to the equation.

1. The result of Andre’s first step was $\text-x+6=\text-8$.
2. The result of Diego’s first step was $6=x-8$.

1. Complete the table with values for $x$ or $y$ that make this equation true: $3x+y=15$.

   $x$	2		6	0	3
   $y$		3				0	8

2. Create a graph, plot these points, and find the slope of the line that goes through them.

Mai and Tyler work on the equation $\frac25b+1=\text-11$ together. Mai's solution is $b=\text-25$ and Tyler's is $b=\text-28$. Here is their work:

Do you agree with their solutions? Explain or show your reasoning.

Solve $3(x-4)=12x$

Describe what is being done in each step while solving the equation.

1. $2(\text-3x+4)=5x+2$
2. $\text-6x+8=5x+2$
3. $8=11x+2$
4. $6=11x$
5. $x=\frac{6}{11}$

A length of ribbon is cut into two pieces to use in a craft project. The graph shows the length of the second piece, $x$, for each length of the first piece, $y$.

1. How long is the ribbon? Explain how you know.
2. What is the slope of the line?
3. Explain what the slope of the line represents and why it fits the story.

Solve each of these equations. Explain or show your reasoning.

Solve each equation, and check your solution.

A participant in a 21-mile walkathon walks at a steady rate of 3 miles per hour. He thinks, “The relationship between the number of miles left to walk and the number of hours I already walked can be represented by a line with slope $\text-3$.” Do you agree with his claim? Explain your reasoning.

Solve each of these equations. Explain or show your reasoning.

1. $2b+8-5b+3=\text-13+8b-5$
2. $2x+7-5x+8=3(5+6x)-12x$
3. $2c-3=2(6-c)+7c$

Solve each equation and check your solution.

1. $\text-3w-4=w+3$
2. $3(3-3x)=2(x+3)-30$
3. $\frac13(z+4)-6=\frac23(5-z)$

Elena said the equation $9x+15=3x+15$ has no solutions because $9x$ is greater than $3x$. Do you agree with Elena? Explain your reasoning.

The table gives some sample data for two quantities, $x$ and $y$, that are in a proportional relationship.

	$x$	$y$
row 1	14	21
row 2	64
row 3		39
row 4	1

1. Complete the table.
2. Write an equation that represents the relationship between $x$ and $y$ shown in the table.
3. Graph the relationship. Use a scale for the axes that shows all the points in the table.

For each equation, decide if it is always true or never true.

1. $x - 13 = x + 1$

2. $x+\frac{1}{2} = x - \frac{1}{2}$

3. $2(x + 3) = 5x + 6 - 3x$

4. $x - 3 = 2x - 3 -x$

5. $3(x-5) = 2(x-5) + x$

Mai says that the equation $2x + 2 = x +1$ has no solution because the left hand side is double the right hand side. Do you agree with Mai? Explain your reasoning.

1. Write the other side of this equation so it's true for all values of $x$: $\frac12(6x-10) - x =$

2. Write the other side of this equation so it's true for no values of $x$: $\frac12(6x-10) - x = $

Here is an equation that is true for all values of $x$: $5(x+2) = 5x+10$. Elena saw this equation and says she can tell $20(x+2)+31=4(5x+10)+31$ is also true for any value of $x$. How can she tell? Explain your reasoning.

Elena and Lin are trying to solve $\frac12x+3=\frac72x+5$. Describe the change they each make to each side of the equation.

1. Elena’s first step is to write $3=\frac72x-\frac12x+5$.
2. Lin’s first step is to write $x+6=7x+10$.

Solve each equation and check your solution.

The point $(\text-3, 6)$ is on a line with a slope of 4.

1. Find two more points on the line.
2. Write an equation for the line.

Lin was looking at the equation $2x-32+4(3x-2462) = 14x$. She said, “I can tell right away there are no solutions, because on the left side, you will have $2x+12x$ and a bunch of constants, but you have just $14x$ on the right side.” Do you agree with Lin? Explain your reasoning.

Han was looking at the equation $6x-4+2(5x+2)=16x$. He said, “I can tell right away there are no solutions, because on the left side, you will have $6x+10x$ and a bunch of constants, but you have just $16x$ on the right side.” Do you agree with Han? Explain your reasoning.

Decide whether each equation is true for all, one, or no values of $x$.

1. $6x-4=\text-4+6x$
2. $4x-6=4x+3$
3. $\text-2x+4=\text-3x+4$

Solve each of these equations. Explain or show your reasoning.

1. $3(x-5) = 6$

2. $2\left(x - \frac{2}{3}\right) = 0$

3. $4x - 5 = 2 -x$

The points $(\text-2,0)$ and $(0,\text-6)$ are each on the graph of a linear equation. Is $(2,6)$ also on the graph of this linear equation? Explain your reasoning.

In the picture triangle $A’B’C’$ is an image of triangle $ABC$ after a rotation. The center of rotation is $E$.

1. What is the length of side $AB$? Explain how you know.
2. What is the measure of angle $D'$? Explain how you know.

Cell phone Plan A costs \$70 per month and comes with a free \$500 phone. Cell phone Plan B costs \$50 per month but does not come with a phone. If you buy the \$500 phone and choose Plan B, how many months is it until your cost is the same as Plan A's?

Priya and Han are biking in the same direction on the same path.

1. Han is riding at a constant speed of 16 miles per hour. Write an expression that shows how many miles Han has gone after $t$ hours.

2. Priya started riding a half hour before Han. If Han has been riding for $t$ hours, how long has Priya been riding?

3. Priya is riding at a constant speed of 12 miles per hour. Write an expression that shows how many miles Priya has gone after Han has been riding for $t$ hours.

4. Use your expressions to find when Han and Priya meet.
    

For what value of $x$ do the expressions $\frac23x+2$ and $\frac43x-6$ have the same value?

Decide whether each equation is true for all, one, or no values of $x$.

1. $2x+8=\text-3.5x+19$
2. $9(x-2)=7x+5$
3. $3(3x+2)-2x=7x+6$

Solve each equation. Explain your reasoning.

1. \(3d+16 = \text-2(5-3d)\)
2. \(2k-3(4-k)=3k+4\)
3. \(\frac{3y-6}{9}=\frac{4-2y}{\text-3}\)

Describe a rigid transformation that takes Polygon A to Polygon B.

1. Match the lines $m$ and $n$ to the statements they represent:
   

   1. A set of points where the coordinates of each point have a sum of 2

   2. A set of points where the $y$-coordinate of each point is 10 less than its $x$-coordinate

2. Match the labeled points on the graph to statements about their coordinates:

   1. Two numbers with a sum of 2

   2. Two numbers where the $y$-coordinate is 10 less than the $x$-coordinate

   3. Two numbers with a sum of 2 and where the $y$-coordinate is 10 less than the $x$-coordinate

Here is an equation: $4x-4=4x+\text{__}$. What could you write in the blank so the equation would be true for:

1. No values of $x$
2. All values of $x$
3. One value of $x$

Mai earns \$7 per hour mowing her neighbors' lawns. She also earned $14 for hauling away bags of recyclables for some neighbors.

1. How many hours do they have to work before they go to the movies?

2. How much will they have earned?

3. Explain where the solution can be seen in tables of values, graphs, and equations that represent Priya's and Mai's hourly earnings.

For each equation, explain what you could do first to each side of the equation so that there would be no fractions. You do not have to solve the equations (unless you want more practice).

Diego has \$11 and begins saving \$5 each week toward buying a new phone. At the same time that Diego begins saving, Lin has \$60 and begins spending \$2 per week on supplies for her art class. Is there a week when they have the same amount of money? How much do they have at that time?

Use a graph to find $x$ and $y$ values that make both $y=\frac {\text{-}2}{3}x+3$ and $y=2x-5$ true.

The point where the graphs of two equations intersect has $y$-coordinate 2. One equation is $y=\text-3x+5$. Find the other equation if its graph has a slope of 1.

A farm has chickens and cows. All the cows have 4 legs and all the chickens have 2 legs. All together, there are 82 cow and chicken legs on the farm. Complete the table to show some possible combinations of chickens and cows to get 82 total legs.

	number of chickens ($x$)	number of cows ($y$)
row 1	35
row 2	7
row 3		10
row 4	19
row 5		5

Here is a graph that shows possible combinations of chickens and cows that add up to 30 animals: If the farm has 30 chickens and cows, and there are 82 chicken and cow legs all together, then how many chickens and how many cows could the farm have?

Here is the graph for one equation in a system of equations:

1. Write a second equation for the system so it has infinitely many solutions.
2. Write a second equation whose graph goes through $(0,1)$ so the system has no solutions.
3. Write a second equation whose graph goes through $(0,2)$ so the system has one solution at $(4,1)$.

Create a second equation so the system has no solutions.

\(\begin{cases} y=\frac34x -4 \\ \\ \end{cases}\)

Andre is in charge of cooking broccoli and zucchini for a large group. He has to spend all \$17 he has and can carry 10 pounds of veggies. Zucchini costs \$1.50 per pound and broccoli costs \$2 per pound. One graph shows combinations of zucchini and broccoli that weigh 10 pounds and the other shows combinations of zucchini and broccoli that cost \$17.

1. Name one combination of veggies that weighs 10 pounds but does not cost \$17.
2. Name one combination of veggies that costs \$17 but does not weigh 10 pounds.
3. How many pounds each of zucchini and broccoli can Andre get so that he spends all \$17 and gets 10 pounds of veggies?

The temperature in degrees Fahrenheit, $F$, is related to the temperature in degrees Celsius, $C$, by the equation $$F = \frac{9}{5}C + 32$$

1. In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?

2. In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?

3. There is one temperature where the degrees Fahrenheit and degrees Celsius are the same, so that $C=F$. Use the expression from the equation, where $F$ is expressed in terms of $C$, to solve for this temperature.

1. Write equations for the lines shown.

   

2. Describe how to find the solution to the corresponding system by looking at the graph.
    

3. Describe how to find the solution to the corresponding system by using the equations.
    

Solve the system of equations: \(\begin{cases} y=4x-3 \\ y=\text-2x+9 \\ \end{cases}\)

Solve the system of equations: \(\begin{cases} y=\frac54x-2 \\ y= \frac {\text{-}1}{4}x+19 \\ \end{cases}\)

Here is an equation: $\frac{15(x-3)}{5}= 3(2x-3)$

1. Solve the equation by using the distributive property first.
2. Solve the equation without using the distributive property.
3. Check your solution.

Solve: \(\begin{cases} y=6x \\ 4x+y=7 \\ \end{cases}\)

Solve: \(\begin{cases} y=3x \\ x=\text-2y+70 \\ \end{cases}\)

The system $x-6y=4$, $3x-18y=4$ has no solution.

1. Change one constant or coefficient to make a new system with one solution.

2. Change one constant or coefficient to make a new system with an infinite number of solutions.

Kiran and his cousin work during the summer for a landscaping company. Kiran's cousin has been working for the company longer, so his pay is 30% more than Kiran's. Last week his cousin worked 27 hours, and Kiran worked 23 hours. Together, they earned \$493.85. What is Kiran's hourly pay? Explain or show your reasoning.

Decide which story can be represented by the system of equations $y=x+6$ and $x+y=100$. Explain your reasoning.

1. Diego’s teacher writes a test worth 100 points. There are 6 more multiple choice questions than short answer questions.
2. Lin and her younger cousin measure their heights. They notice that Lin is 6 inches taller, and their heights add up to exactly 100 inches.
    

Clare and Noah play a game in which they earn the same number of points for each goal and lose the same number of points for each penalty. Clare makes 6 goals and 3 penalties, ending the game with 6 points. Noah earns 8 goals and 9 penalties and ends the game with $\text-22$ points.

1. Write a system of equations that describes Clare and Noah's outcomes. Use $x$ to represent the number of points for a goal and $y$ to represent the number of points for a penalty.

2. Solve the system. What does your solution mean?

Solve: \(\begin{cases} y=6x-8 \\ y=\text-3x+10 \\ \end{cases}\)

1. Estimate the coordinates of the point where the two lines meet.

   

2. Choose two equations that make up the system represented by the graph.

   1. $y=\frac54x$

   2. $y=6-2.5x$

   3. $y=2.5x+6$

   4. $y=6-3x$

   5. $y=0.8x$

3. Solve the system of equations and confirm the accuracy of your estimate.