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Questions and Answers

What is the value of a in the equation $3a+b=54$, when $b=9$?

15 (correct)

The perimeter of the scalene triangle is 54.6 cm. Which equation can be used to find the value of b if side a measures 8.7 cm?

8.7+b=54.6

17.4+b=54.6

26.1+b=54.6

34.8+b=54.6 (correct)

What type of assessment is 'Writing and Graphing Equations in Two Variables' referring to?

Pre-test / Quiz

For each new desk that is purchased, by how much does the amount of money left in the school's desk fund decrease?

$110 Signup and view all the answers

Which graph represents viable values for $y = 2x$, where x is the number of pounds of rice scooped and purchased from a bulk bin at the grocery store?

Graph A Signup and view all the answers

Peaches are being sold for $2 per pound. Which statement best describes the values of x and y?

The values of both x and y will be real numbers greater than or equal to 0. Signup and view all the answers

What is the equation used to find the total amount of flour that Otto used in the recipe?

y=x+6; x is any real number greater than or equal to 0, and y is any real number greater than or equal to 6. Signup and view all the answers

Which graph represents viable values for $y = 5.5x$, where x is the number of cans of tomato paste and y is the total weight of the sealed cans in ounces?

Graph B Signup and view all the answers

Jayne stopped to get gas. Which equation relates the total amount of gasoline in the tank, y, to the number of gallons that she added, x?

y=4+x Signup and view all the answers

Which graph shows the equation $V = 4 + 2t$, where V is the total volume of water in a bucket and t is the elapsed time in minutes?

Graph C Signup and view all the answers

Which description is represented by a discrete graph?

Kiley bought a platter for $19 and several matching bowls that were $8 each. What is the total cost before tax? Signup and view all the answers

Match the following ordered pairs with their context:

(-3, -0.90) = Represents a negative delay in days and fee. (-2.5, -0.75) = Represents a non-integer late day and fee. (4.5, 1.35) = Represents late days and a fee. (8, 2.40) = Represents days overdue and their fee. Signup and view all the answers

The graph shows the relationship between the total cost and the number of gift cards that Raj bought for raffle prizes. What would be the cost for 5 of the gift cards?

$100 Signup and view all the answers

Which ordered pair is a viable solution if x represents the number of books he orders and y represents the total weight of the books, in ounces?

(0, 0) Signup and view all the answers

Which scenario is most likely the one shown on the graph?

The total number of calories, y, in a salad with vegetables containing 50 calories topped with x ounces of salad dressing at 100 calories per ounce. Signup and view all the answers

The graph shows the relationship between the number of hours that Michelle has been driving and the distance that she has left to travel to get to her destination. Which statement is true?

For each hour that Michelle drove, she traveled an additional 50 miles. Signup and view all the answers

What is the equation that can be used to determine the total length of all of the yarn that Julie ends up cutting?

y=4x+7.75; continuous Signup and view all the answers

What is the range of water required for cooking rice?

All real numbers such that 0 ≤ y ≤ 40 Signup and view all the answers

What is the domain of the function on the graph?

all real numbers greater than or equal to -3 Signup and view all the answers

What is the lowest value of the range of the function shown on the graph?

-2 Signup and view all the answers

Which explains why the graph is not a function?

It is not a function because there are two different x-values for a single y-value. Signup and view all the answers

What is the range of the function on the graph?

all real numbers less than or equal to 3 Signup and view all the answers

Match the following sets with their definitions:

{x | x = -5, -3, 1, 2, 6} = What is the domain of the given function? {y | y = -7, -1, 0, 9} = What is the range of the given function? {x | x = -9, -6, -5, -3, 0, 1, 2, 4, 6} = What is the domain of the function shown in the mapping? {y | y = -9, -6, 0, 2, 4} = What is the range of the given function? Signup and view all the answers

What does the word 'Subject' represent?

Representing relationships Signup and view all the answers

What is the best description of Quantitative Reasoning?

Pre-test / Quiz Signup and view all the answers

Which table could represent the speed of a car that increases, then decreases?

Table B Signup and view all the answers

Which statement best describes the heart rate during exercise?

The heart rate increases for 6 minutes, remains constant for 19 minutes, and then gradually decreases for 5 minutes. Signup and view all the answers

What statement can be supported by the table showing the tree height over 11 months?

The tree increased in height each month until the period between month 9 and month 11. Signup and view all the answers

How does the dog's movement 5 seconds after the command compare?

The dog was running towards the trainer. Signup and view all the answers

Which statement is true about the climbers' heights?

Both climbers rested on the wall at a constant height for 2 minutes. Signup and view all the answers

How high did Abby climb above her original starting position?

18 feet Signup and view all the answers

During which interval did the average time spent on hold increase?

Week 4 to Week 6 Signup and view all the answers

Which graph could represent a constant balance in a bank account over time?

Graph C Signup and view all the answers

Which statements are true about the temperature recorded from 8:00 a.m. to 8:00 p.m.?

The temperature increased until 4:00 p.m. Signup and view all the answers

Which statement best describes the relationship between storage space and the number of music files?

As the number of files increases, storage space used increases. Signup and view all the answers

Which statement best describes the speed situation in section C of Sam's car?

His car is slowing down. Signup and view all the answers

What could describe Yana’s activities in the computer lab?

Yana opened her saved paper, wrote, took a break, deleted some of the words, and then began writing again. Signup and view all the answers

How many minutes did Lynn run at a greater speed than Kael?

28 Signup and view all the answers

How many days overdue could Christen's overdue book fine be?

22 Signup and view all the answers

What describes the pilot's distance in section I?

Decreasing Signup and view all the answers

What is happening to Jamal's speed in section B?

Jamal is traveling at a constant speed. Signup and view all the answers

During which one-day interval did Katrina work the most hours?

From day 7 to day 8. Signup and view all the answers

At what times could Rory's phone have been plugged into the charger?

3:00 p.m. Signup and view all the answers

Which sets of values could represent Amber's exercise intensities during the fourth, fifth, and sixth visits?

66%, 69%, 72% Signup and view all the answers

What describes Myra's distance as time increases?

Increasing Signup and view all the answers

How does y relate to x?

As x increases, y decreases and then increases. Signup and view all the answers

What is the definition of 'Dimensional Analysis'?

Pre-test / Quiz. Signup and view all the answers

What is the value of m when the equation is x + 5 = 35?

44 Signup and view all the answers

What is the approximate price of a pint of veggies if fruit costs $2.85 and total spending is $28.70?

$1.75 Signup and view all the answers

What is the value of x in the equation x - y = 30 when y = 15?

200 Signup and view all the answers

What is the value of y in the equation 6.4x + 2.8y = 44.4 when x = 3?

y = 9 Signup and view all the answers

What error did Malik make when solving the equation?

Malik substituted 60 for y instead of x. Signup and view all the answers

Which equation models Hanna's purchases with $43.92?

2.99x + 12.99y = 43.92 Signup and view all the answers

Which equation can be used to find the number of times Maria hiked each trail?

90 - 10y = 5x Signup and view all the answers

What equation can find the value of x if the isosceles triangle has a perimeter of 7.5 m?

2.1 + 2x = 7.5 Signup and view all the answers

What is the value of y when x = 0.3 in 5x + 2y = 20?

9.25 Signup and view all the answers

What is the value of x when y = 4 in 8x - 2y = 48?

7 Signup and view all the answers

What error did Barbara make in writing her equation?

Barbara's equation did not consider the number of bottles of iced tea. Signup and view all the answers

What values of x and y could represent the number of DVDs purchased?

x = 16, y = 22 Signup and view all the answers

Which shows the correct solution of the equation 1/2a + 2/3b = 50 when b = 30?

C Signup and view all the answers

Which equation models Chin's canning scenario if he canned a total of 1,280 ounces?

16p + 32q = 1280 Signup and view all the answers

Study Notes

Fundamental Concepts in Algebra I

Subject: Represents relationships within mathematical contexts, essential for understanding variables and functions.
Quantitative Reasoning: Involves analyzing numerical information to solve problems, foundational for higher-level math.

Relationships and Graphs

A car's speed fluctuations can be represented in a table format, indicating acceleration and deceleration.
Heart rate during exercise typically shows an initial increase, followed by a steady state, and then potentially a decrease, illustrating physiological responses.
Graphical representations of climbers' heights reveal both climbers could rest at constant heights, emphasizing the importance of steady performance in activities like climbing.

Data Representation

Trees typically exhibit growth patterns over time, where specific months may mark growth plateaus.
Distance between a dog and trainer can be tracked over time, reflecting the dog's movement dynamics.
Temperature changes over the day can create patterns observed in graphs, relevant for predicting seasonal variations.

Time Intervals and Changes

Changes in patterns over time can be evidenced by examining averages, such as hold times on calls, indicating service quality.
Consistent graph representation helps understand financial data, like bank balances remaining unchanged over time.

Speed and Motion

Speed analyses indicate when a vehicle is slowing down or maintaining steady speeds, critical for assessments of vehicle performance.
Minute intervals of speed in physical activities, such as running track, help conclude overall performance metrics.

Equations and Mathematical Relationships

Relationships involving variables can be formulated into equations to model scenarios (e.g., hiking trails).
Common errors in equation solving often stem from incorrect substitutions or misinterpretations of variable relationships.

Unit Conversions and Measurements

Conversions between different units are vital; for example: inches to feet and miles to kilometers.
Understanding weight metrics (kilograms to pounds) and area measurements is necessary for practical applications in various fields.

Problem-Solving Techniques

Applications of dimensional analysis facilitate performing unit conversions for scientific and engineering purposes.
Identifying equations that correctly model real-world scenarios is crucial for problem-solving; errors can occur from not considering all necessary variables.

Financial Understanding

Cost comparisons in a grocery setting illustrate the importance of calculating unit prices for budget-conscious decisions.
Awareness of constraints while purchasing items can help individuals make informed decisions within budgetary limits.

Summary Formulation

The ability to model real-world situations mathematically is crucial in fields ranging from economics to engineering.
Understanding variables, their relationships, and applying mathematical reasoning to graph interpretations forms the basis of efficient problem-solving.### Equations and Variations
y = 6x: Represents linear functions where x is a non-negative integer or real number; y starts from 6 and increases in steps of 6.
y = x + 6: Indicates another linear function applicable for both integer and real values of x, with y starting from 6.
Definition Extended: The definition of y = x + 6 is reiterated, confirming its applicability for all real numbers.

Viable Values and Scenarios

Graph Representation: Graph B shows valid outputs for y = 5.5x, where x signifies the number of tomato paste cans.
Gasoline Equation: Total gasoline y = 4 + x describes Jayne's tank starting with 4 gallons and increasing as she adds more.
Volume Equation: Graph C corresponds with the equation V = 4 + 2t, modeling water volume over time.

Discrete vs Continuous Scenarios

Discrete Graph Example: Kiley's total platter and bowls cost scores are discrete, as she buys whole bowls (not fractions).
Library Fee Calculation: The ordered pair (8, 2.40) shows a realistic solution for late fees based on integer days late.
Cost Estimation of Gift Cards: The total cost for 5 gift cards is determined to be $100 based on prior purchasing behavior.

Parameters and Constraints

Weight Calculation for Books: (0, 0) is a valid solution, indicating no weight for zero books ordered.
Calories in Salad: Quantifies total calories combining vegetable salad and dressing based on consistent unit pricing.
Distance and Time Relation: Acknowledges that between distance traveled and time spent driving (50 miles for each hour).

Functions and Relationships

Homework Time Equation: y = 2x + 20, where total time spent increases with integers of x math problems solved.
Jogging Speed: Maria's jogging speed is calculated at 0.1 miles per minute based on the provided graph.
Bread Length Over Time: The relationship l = 65 - 15d tracks the length of bread after daily consumption.

Graph Characteristics

Continuous Graph For Gas: A continuous representation of the gas tank’s capacity change due to possible non-integer values.
Yarn Length Equation: All pieces of yarn with 7.75 inches yield a continuous sum, expressed through t = 4y + 7.75.

Cost Models

Notebook Pricing: Each notebook costs $3, with integer values for x (number of notebooks) impacting total cost y.
Carnival Cost: Graph B relates to total spending for carnival tickets based on variable purchases.

Membership Increase in Clubs

Discrete Membership Growth: Yearbook club growth over days demonstrates a discrete increase, with individuals added each time.

Fundamental Function Concepts

Range and Domain Discussion: Frequently emphasizes understanding the lowest values, dependent and independent variables, suitability, and distinguishing features of functions, such as ordered pairs from graphs.

Specific Data Evaluations

Water Usage and Units: u = 748g identifies g (gallons used) as independent, while u (units) becomes dependent.
Bathtub Water Remaining: Model tracks water draining with a maximum threshold described.

Functional Validity and Adjustments

Assessing Valid Functions: Identification of non-function statuses based on multiple y-values for a single x-value reinforces function definitions.
Necessary Graph Adjustments: Understanding how many points need removal to uphold function criteria solidifies core graph theory.

Conclusion and Summary Elements

Critical Review of Graphs and Functions: Comprehensive evaluation of various cases and equations offers valuable insights into linear functions, independent vs. dependent variables, and the characteristics of continuous versus discrete data.

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