Lesson 9.3Write Equations to Represent Relationships Between Two Variables
Start hereWords, worked example, and sentence starters
Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| equationSpanish: ecuación | A mathematical sentence with an equals sign that shows two amounts are the same. | y = 5x |
| variableSpanish: variable | A letter that represents a quantity that can change, or vary. | x, y, c, h |
| rateSpanish: tasa | A fixed amount of one quantity for every one unit of the other quantity. | $3 for each pound |
| representSpanish: representar | To show a relationship using a tool such as a table, a graph, or an equation. | The same membership shown as a table, a graph, AND c = 24.95m. |
| define the variablesSpanish: definir las variables | To state exactly what quantity each letter in an equation stands for. | c = total cost |
| ConstantSpanish: Constante | A number in an equation that never changes. | the 5 in y = 5x |
How it worksWorked examplewriting an equation from a table
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A table shows x: 1, 2, 3, 4 hours worked and y: 18, 36, 54, 72 dollars earned. Write an equation.
- Define the variables: x = hours worked, y = total dollars earned.
- Check each row for one rate: 18 ÷ 1, 36 ÷ 2, and 54 ÷ 3 all give 18.
- The constant rate is 18 dollars per hour, so y is 18 times x.
- Write it: y = 18x. Test the last row: 18 × 4 → 72, which matches.
Answer: y = 18x, where x is hours worked and y is total dollars earned.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A table shows x: 1, 2, 3, 4 bags of sand and y: 11, 22, 33, 44 pounds. Write an equation.
- Define the variables: x = , y = .
- Check each row for one rate: ÷ 1, ÷ 2, ÷ 3 all give .
- The constant rate is pounds per bag, so y is times x.
- Write it: y = x. Then test the last row of the table to check.
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- In my equation, x stands for and y stands for .
- The constant rate is because every y value is .
- My equation is y = , which means .
- I checked my equation by substituting for x.
Word bank equationvariablerateconstantdefinesubstitutetimespertablerowrelatecheck
Watch outA common mistake
Writing the equation with addition — c = 24.95 + m — instead of multiplication. The fee is charged EVERY month, so the total is the rate times the number of months; always test the equation against a row of the table.
Lesson 9.3Write Equations to Represent Relationships Between Two Variables
Version ASupported practice
Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
-
1Circle the letter of the best answer. Show how you know.
In the equation c = 24.95m for the gym membership, what does m represent?
- AThe number of months of membership
- BThe total cost in dollars
- CThe monthly fee of $24.95
- DThe name of the gym
Hint Which letters are variables here — and which quantity does each stand for?
Show your work
-
2Circle the letter of the best answer. Show how you know.
Recreation World charges $24.95 per month. What is the total cost of 2 months of membership?
- A$24.95
- B$26.95
- C$49.90
- D$74.85
Hint What does ONE month cost?
per 1 - 1LabelWrite what each column is.
- 2Find per 1Divide to get one.
- 3ScaleMultiply up to what is asked.
- 4CheckDoes the size make sense?
-
3Circle the letter of the best answer. Show how you know.
A canoe rents for $15 per hour. Let h = hours and c = total cost. Which equation represents the relationship?
- Ac = 15h
- Bc = 15 + h
- Ch = 15c
- Dc = h − 15
Hint Say it in words: the total cost equals $15 ___ the number of hours.
Show your work -
4Circle the letter of the best answer. Show how you know.
Using c = 15h, what is the total cost of a 3-hour canoe rental?
- A$15
- B$18
- C$45
- D$75
Hint Replace h with 3 in the equation.
My work — one step per line What I did to both sides - 1LookWhat is done to the variable?
- 2UndoDo the opposite — both sides.
- 3SimplifyOne side at a time.
- 4CheckPut it back in.
-
5Circle the letter of the best answer. Show how you know.
In the gym membership situation, which quantity is the DEPENDENT variable?
- AThe number of months
- BThe total cost
- CThe monthly fee of $24.95
- DThe number of gyms in town
Hint Which quantity responds to the other?
Show your work -
6Circle the letter of the best answer. Show how you know.
The membership table shows: 1 month, $24.95; 2 months, $49.90; 3 months, $74.85. By how much does the total cost increase each month?
- A$24.05
- B$24.95
- C$25.00
- D$49.90
Hint Subtract one row's total from the next row's total.
Show your work
Explain your thinkingBoth equations charge by a rate. What do m and h each stand for in their own situation, and why would c = 24.95m be wrong for the canoe?
Sentence starter In ___, ___ stands for ___, so the rate ___ means ___.
Lesson 9.3Write Equations to Represent Relationships Between Two Variables
Version BCore practice
Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
-
1Circle the letter of the best answer. Show how you know.
A table pairs months with total cost: (1, 24.95), (2, 49.90), (4, 99.80). Which equation matches the table, with m = months and c = cost?
- Ac = 24.95m
- Bc = m + 24.95
- Cm = 24.95 + c
- Dc = 49.90m
Show your work
-
2Circle the letter of the best answer. Show how you know.
Using c = 24.95m, what is the total cost of 10 months of membership?
- A$34.95
- B$224.55
- C$249.50
- D$2,495.00
My work — one step per line What I did to both sides -
3Circle the letter of the best answer. Show how you know.
Using c = 15h, what is the total cost of a 5-hour canoe rental?
- A$20
- B$60
- C$75
- D$80
My work — one step per line What I did to both sides -
4Circle the letter of the best answer. Show how you know.
On the canoe graph with hours on the horizontal axis and cost on the vertical axis, which point lies on the graph?
- A(4, 60)
- B(4, 19)
- C(4, 55)
- D(60, 4)
Show your work -
5Circle the letter of the best answer. Show how you know.
The friends have $90 to spend. Using c = 15h, for how many hours can they rent the canoe?
- A5 hours
- B6 hours
- C75 hours
- D1,350 hours
My work — one step per line What I did to both sides -
6Circle the letter of the best answer. Show how you know.
How are the canoe graph and the equation c = 15h related?
- AThe graph and the equation show different relationships
- BThe equation only works for the points shown on the graph
- CEvery point on the graph is a pair (h, c) that makes the equation true
- DThe graph replaces the equation, so only one can be used
Show your work
Explain your thinkingBoth equations charge by a rate. What do m and h each stand for in their own situation, and why would c = 24.95m be wrong for the canoe?
Lesson 9.3Write Equations to Represent Relationships Between Two Variables
ChallengeExtension
Learning target I can write an equation from a table or a graph to model the relationship between the dependent and independent variables.
-
1Write the letter of the matching item on each line.
Match each scenario to the equation that models it.
- Canoe rental: $15 per hour
- Gym membership: $24.95 per month
- Football tickets: $45 each
- Tram climbing 1,200 feet per minute
- Ac = 15h
- Bc = 24.95m
- Cc = 45t
- Dd = 1200m
-
2Circle the letter of the best answer. Show how you know.
Miguel wants the cost of a full year (12 months) at Recreation World. Using c = 24.95m, what is the total?
- A$36.95
- B$249.50
- C$299.40
- D$300.00
My work — one step per line What I did to both sides -
3Circle the letter of the best answer. Show how you know.
A different gym charges $19.95 per month. How does Miguel's equation change for this gym?
- AThe variable changes: c = 24.95x instead of c = 24.95m
- BThe rate changes: c = 19.95m instead of c = 24.95m
- CThe operation changes: c = 19.95 + m
- DNothing changes — all gyms have the same equation
Show your work -
4Circle the letter of the best answer. Show how you know.
What is the benefit of an equation over a table for the membership costs?
- AThe equation lists more rows than the table
- BThe equation is the only representation that shows the cost increasing
- CThe table is always wrong and the equation is always right
- DThe equation gives the total cost for ANY number of months, not just the rows listed
Show your work -
5Circle the letter of the best answer. Show how you know.
The friends spent exactly $120 on the canoe. Solve 15h = 120. How long was the rental?
- A8 hours
- B9 hours
- C105 hours
- D1,800 hours
My work — one step per line What I did to both sides