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Key Questions

extrapolate the data in a linear manner to predicate a value

Explanation:

For example making a graph of temperature vs volume can be used to predict a value for absolute zero. The linear line can be extended to find a predicted value for the temperature where there is zero volume. The place on the line where volume is zero would be predict value for absolute zero.

problem solving with linear models iready answers

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2.5.4: Applications Using Linear Models

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Applications Using Linear Models

Suppose a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. Could you write a linear equation to model this situation? Would slope-intercept form, point-slope form, or standard form be easiest to use?

Applying Linear Models

Modeling linear relationships can help solve real-world applications. Consider the example situations below, and note how different problem-solving methods may be used in each.

  • Nadia has $200 in her savings account. She gets a job that pays $7.50 per hour and she deposits all her earnings in her savings account. Write the equation describing this problem in slope-intercept form. How many hours would Nadia need to work to have $500 in her account?

problem solving with linear models iready answers

Begin by defining the variables:

y= amount of money in Nadia’s savings account

x= number of hours

The y-intercept ($200) and the slope of the equation ($7.501 hour) are given.

We are told that Nadia has $200 in her savings account, so b=200.

We are told that Nadia has a job that pays $7.50 per hour, so m=7.50.

By substituting these values into slope–intercept form, y=mx+b, we obtain y=7.5x+200.

To answer the question, substitute $500 for the value of y and solve.

500=7.5x+200

Nadia must work 40 hours if she is to have $500 in her account.

  • Marciel rented a moving truck for the day. Marciel remembers only that the rental truck company charges $40 per day and some amount of cents per mile. Marciel drives 46 miles and the final amount of the bill (before tax) is $63. What is the amount per mile the truck rental company charges? Write an equation in point-slope form that describes this situation. How much would it cost to rent this truck if Marciel drove 220 miles?

problem solving with linear models iready answers

Define the variables: x= distance in miles; y= cost of the rental truck in dollars. There are two ordered pairs: (0, 40) and (46, 63).

Step 1: Begin by finding the slope: 63−4046−0=2346=12

Step 2: Substitute the slope for m and one of the coordinates for (x1,y1).

y−40=12(x−0)

To find out how much will it cost to rent the truck for 220 miles, substitute 220 for the variable x.

y−40=1/2(220−0)

y−40=0.5(220)

  • Nimitha buys fruit at her local farmer’s market. This Saturday, oranges cost $2 per pound and cherries cost $3 per pound. She has $12 to spend on fruit. Write an equation in standard form that describes this situation. If she buys 4 pounds of oranges, how many pounds of cherries can she buy?

problem solving with linear models iready answers

Define the variables: x= pounds of oranges and y= pounds of cherries.

The equation that describes this situation is: 2x+3y=12

If she buys 4 pounds of oranges, we substitute x=4 into the equation and solve for y.

3y=12−8

Nimitha can buy 1(1/3) pounds of cherries.

Example 2.5.4.1

Earlier, you were told that a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. What linear equation would model this situation?

In this example, you are given the slope of the line that would represent this situation: 3 (because each rental costs $3.00). You are also given the point (8, 30) because when you rent 8 movies, your bill is $30.00. So, you have the slope and a point. This means that the best form to use to write an equation is point-slope form.

To write the equation, first define the variables: x= number of movies rented; y= the monthly bill in dollars. The slope is 3 and one ordered pair is (8, 30).

Since you have the slope, substitute the slope for m and the coordinate for (x1,y1) into the point-slope form equation:

y−30=3(x−8)

You can rewrite this in slope-intercept form by using the Distributive Property and the Addition Property of Equality:

y−30=3x−24

So the equation that models this situation is y−30=3(x−8) or y=3x+6.

Example 2.5.4.2

A stalk of bamboo of the family Phyllostachys nigra grows at steady rate of 12 inches per day and achieves its full height of 720 inches in 60 days. Write the equation describing this problem in slope-intercept form. How tall is the bamboo 12 days after it started growing?

Define the variables.

y= the height of the bamboo plant in inches

x= number of days

The problem gives the slope of the equation and a point on the line.

The bamboo grows at a rate of 12 inches per day, so m=12.

We are told that the plant grows to 720 inches in 60 days, so we have the point (60, 720).

Start with the slope-intercept form of the line. y=mx+b

Substitute 12 for the slope. y=12x+b

Substitute the point (60,720). 720=12(60)+b ⇒b=0

Substitute the value of b back into the equation. y=12x

To answer the question, substitute the value x=12 to obtain y=12(12)=144 inches.

The bamboo is 144 inches 12 days after it starts growing.

Example 2.5.4.3

Jethro skateboards part of the way to school and walks for the rest of the way. He can skateboard at 7 miles per hour and he can walk at 3 miles per hour. The distance to school is 6 miles. Write an equation in standard form that describes this situation. If Jethro skateboards for 12 of an hour, how long does he need to walk to get to school?

problem solving with linear models iready answers

Define the variables: x= hours Jethro skateboards and y= hours Jethro walks.

The equation that describes this situation is 7x+3y=6.

If Jethro skateboards 12 of an hour, we substitute x=0.5 into the equation and solve for y.

7(0.5)+3y=6

3y=6−3.5

Jethro must walk 5/6 of an hour to get to school.

  • To buy a car, Andrew puts in a down payment of $1500 and pays $350 per month in installments. Write an equation describing this problem in slope-intercept form. How much money has Andrew paid at the end of one year?
  • Anne transplants a rose seedling in her garden. She wants to track the growth of the rose, so she measures its height every week. In the third week, she finds that the rose is 10 inches tall and in the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, write an equation describing this problem in slope-intercept form. What was the height of the rose when Anne planted it?
  • Ravi hangs from a giant exercise spring whose length is 5 m. When his child Nimi hangs from the spring, his length is 2 m. Ravi weighs 160 lbs. and Nimi weighs 40 lbs. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs., hangs from it?
  • Petra is testing a bungee cord. She ties one end of the bungee cord to the top of a bridge and to the other end she ties different weights. She then measures how far the bungee stretches. She finds that for a weight of 100 lbs., the bungee stretches to 265 feet and for a weight of 120 lbs., the bungee stretches to 275 feet. Physics tells us that in a certain range of values, including the ones given here, the amount of stretch is a linear function of the weight. Write the equation describing this problem in slope-intercept form. What should we expect the stretched length of the cord to be for a weight of 150 lbs?
  • Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is 20 cm and for a 300 gram weight the length of the stretched spring is 25 cm. Write an equation in point-slope form that describes this situation. What is the unstretched length of the spring?
  • Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet. Write an equation in point-slope form that describes this situation. What was the submarine’s depth five minutes after it started surfacing?
  • Anne got a job selling window shades. She receives a monthly base salary and a $6 commission for each window shade she sells. At the end of the month, she adds up her sales and she figures out that she sold 200 window shades and made $2500. Write an equation in point-slope form that describes this situation. How much is Anne’s monthly base salary?
  • The farmer’s market sells tomatoes and corn. Tomatoes are selling for $1.29 per pound and corn is selling for $3.25 per pound. If you buy 6 pounds of tomatoes, how many pounds of corn can you buy if your total spending cash is $11.61?
  • The local church is hosting a Friday night fish fry for Lent. They sell a fried fish dinner for $7.50 and a baked fish dinner for $8.25. The church sold 130 fried fish dinners and took in $2,336.25. How many baked fish dinners were sold?
  • Andrew has two part-time jobs. One pays $6 per hour and the other pays $10 per hour. He wants to make $366 per week. Write an equation in standard form that describes this situation. If he is only allowed to work 15 hours per week at the $10 per hour job, how many hours does he need to work per week at his $6 per hour job in order to achieve his goal?
  • Anne invests money in two accounts. One account returns 5% annual interest and the other returns 7% annual interest. In order not to incur a tax penalty, she can make no more than $400 in interest per year. Write an equation in standard form that describes this problem. If she invests $5000 in the 5% interest account, how much money does she need to invest in the other account?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.6.

Additional Resources

PLIX: Play, Learn, Interact, eXplore: The Perfect Lemonade 1

Activity: Applications Using Linear Models Discussion Questions

Practice: Applications Using Linear Models

Real World Application: Tracking the Storm

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7.2: Modeling with Linear Equations

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In this section, you will learn to use linear functions to model real-world applications

Read the problem carefully. Highlight important information. Identify what each variable represents in the context of the problem.

Example \(\PageIndex{1}\)

It costs $750 to manufacture 25 items, and $1000 to manufacture 50 items. Assuming a linear relationship holds, find the cost equation, and use this equation to predict the cost of 100 items.

We let \(x\) = the number of items manufactured, and let \(y\) = the cost.

Solving this problem is equivalent to finding an equation of a line that passes through the points (25, 750) and (50, 1000).

\[ m = \frac{1000-750}{50-25} = 10 \nonumber \]

Therefore, the partial equation is \(y = 10x + b\)

By substituting one of the points in the equation, we get \(b = 500\)

Therefore, the cost equation is \(y = 10x + 500\). This equation is the linear model for this problem.

Now use the linear model to find the cost of 100 items. Substitute \(x = 100\) in the equation \(y = 10x + 500\)

So the cost is

\[y = 10(100) + 500 = 1500 \nonumber\]

It costs $1500 to manufacture 100 items.

Example \(\PageIndex{2}\)

The freezing temperature of water in Celsius is 0 degrees and in Fahrenheit 32 degrees. And the boiling temperatures of water in Celsius, and Fahrenheit are 100 degrees, and 212 degrees, respectively. Write a conversion equation from Celsius to Fahrenheit and use this equation to convert 30 degrees Celsius into Fahrenheit.

Let us look at what is given.

We let \(C\) = the degrees in Celsius, and let \(F\) = the degrees in Fahrenheit.

Again, solving this problem is equivalent to finding an equation of a line that passes through the points (0, 32) and (100, 212).

Since we are finding a linear relationship, we are looking for an equation \(y = mx + b\), or in this case \(F = mC + b\), where \(x\) or \(C\) represent the temperature in Celsius, and y or F the temperature in Fahrenheit.

\[ \text{slope m } = \frac{312-32}{100-0} = \frac{9}{5} \nonumber \]

The equation is \(F = \frac{9}{5}C + b\)

Substituting the point (0, 32), we get b = 32 and the conversion equation is

\[F = \frac{9}{5}C + 32 \nonumber.\]

To convert 30 degrees Celsius into Fahrenheit, substitute \(C = 30\) in the equation

\begin{aligned} &\mathrm{F}=\frac{9}{5} \mathrm{C}+32\\ &\mathrm{F}=\frac{9}{5}(30)+32=86 \end{aligned}

Thus, 30 degrees Celsius is equal to 86 degrees Fahrenheit.

Example \(\PageIndex{3}\)

The variable cost to manufacture a product is $10 per item and the fixed cost $2500. If \(x\) represents the number of items manufactured and \(y\) represents the total cost, write the cost function.

  • The variable cost of $10 per item tells us that \(m = 10\).
  • The fixed cost represents the \(y\)-intercept. So \(b = 2500\).

Therefore, the cost function is \(y = 10x + 2500\).

Example \(\PageIndex{4}\)

Assume a car depreciates by the same amount each year. Joe purchased a car in 2010 for $16,800. In 2014 it is worth $12,000. Find the linear model. Use this model to predict how much the car will be worth in 2020.

We let \(x\) = the number of years after 2010, and let \(y\) = the cost.

Solving this problem is equivalent to finding an equation of a line that passes through the points (0, 16800) and (4, 12000).

To find the linear model for this problem, we need to find the slope.

\[ m = \frac{12000-16800}{4-0} = -1200 \nonumber \]

The slope indicates that the rate of depreciation each year is $-1200. Thus, the linear model for this problem is: \(y = -1200x + 16,800\) Now, to find out how much the car will be worth in 2020, we need to know how many years that is from the purchase year. Since it is ten years later, \(x=10\).

\[y=-1200(10)+16,800=-12,000+16,800=4,800\]

The car will be worth $4800 in 2020.

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Note: The value of the car over time follows a decreasing straight line.

Example \(\PageIndex{5}\)

The cost \(y\), in dollars, of a gym membership for \(n\) months can be described by the linear model \(y=30n+70\). What does this model tell us?

The value for \(y\) when \(n=0\) in this equation is 70, so the initial starting cost is $70. This tells us that there must be an initiation or start-up fee of $70 to join the gym.

The value for the slope, \(m\) in the equation is 30, so the cost increases by $30 each month. This tells us that the monthly membership fee for the gym is $30 a month.

Example \(\PageIndex{6}\)

The population of Canada in the year 1980 was 24.5 million, and in the year 2010 it was 34 million. The population of Canada over that time period can be approximately modeled by a linear function. Let x represent time as the number of years after 1980 and let y represent the size of the population.

  • Write the linear function that gives a relationship between the time and the population.
  • Assuming the population continues to grow linearly in the future, use this equation to predict the population of Canada in the year 2025.

The problem can be made easier by using 1980 as the base year, that is, we choose the year 1980 as the year zero. This will mean that the year 2010 will correspond to year 30. Now we look at the information we have:

a. Solving this problem is equivalent to finding an equation of a line that passes through the points (0, 24.5) and (30, 34). We use these two points to find the slope:

\[ m = \frac{34-24.5}{30-0}=\frac{9.5}{30} = 0.32 \nonumber\]

The \(y\)-intercept occurs when \(x = 0\), so \(b = 24.5\). We write the linear model

\[ y =0.32x + 24.5 \nonumber \]

b. Now to predict the population in the year 2025, we let \(x=2025-1980=45\)

\begin{aligned} &y=0.32 x+24.5\\ &y=0.32(45)+24.5=38.9 \end{aligned}

In the year 2025, we predict that the population of Canada will be 38.9 million people.

Note that we assumed the population trend will continue to be linear. Therefore if population trends change and this assumption does not continue to be true in the future, this prediction may not be accurate.

Definition: Linear Growth

A quantity grows linearly if it grows by a constant amount for each unit of time.

Example \(\PageIndex{7}\): City Growth

Suppose in Flagstaff Arizona, the number of residents increased by 1000 people per year. If the initial population was 46,080 in 1990, can you predict the population in 2013? This is an example of linear growth because the population grows by a constant amount. We list the population in future years below by adding 1000 people for each passing year.

This is the graph of the population growth over a six year period in Flagstaff, Arizona. It is a straight line and can be modeled with a linear growth model.

The population growth, y , can be modeled with a linear equation. The initial population is 46,080. The future population depends on the number of years, t , after the initial year. The model is \(y = 1000t + 46,080\). Note, we chose to use the variable t as a simple reminder that t represents time . We could continue to use the variable x , or any other letter for that matter, but t for time makes sense.

To predict the population in 2013, we identify how many years it has been from 1990 (which is year zero). So t = 23 for the year 2013.

\[y=1000(23)+46,080=69,080\]

The population of Flagstaff in 2013 will be 69,080 people.

Example \(\PageIndex{8}\): Antique Frog Collection

Dora has inherited a collection of 30 antique frogs. Each year she vows to buy two frogs a month to grow the collection. This is an additional 24 frogs per year. How many frogs will she have in six years? How long will it take her to reach 510 frogs?

The initial population is 30 frogs, so b=30. The rate of change is 24 frogs per year, so m=24. The linear growth model for this problem is:

\[y = 24t + 30\] where t = time in years and y = the number of frogs

The first question asks how many frogs will Dora have in six years so, t = 6.

\[y = 24(6)+30 = 144+30 = 174\] frogs.

The second question asks for the time it will take for Dora to collect 510 frogs. So, \(y = 510\)and we will solve for t.

\[\begin{align*}510 &= 24t+30 \\ 480 &= 24t \\ 20 &= t \end{align*}\]

It will take 20 years to collect 510 antique frogs.

G4LXUsPCy-lDKc16lLZcIBeSqMra-0tEXr8PvDPmjH5pdW1BkuF3La9Np-OssuDM5FbcfoC4AcR3L9qNC9fkYsKAwsrye53FvAMt0jFg6HPRNY9viIzJ33HG1pfqY0OELdv7xfU

Note: The graph of the number of antique frogs Dora accumulates over time follows a straight line.

Try it Now 1

The number of stay-at-home fathers in Canada has been growing steadily[1]. While the trend is not perfectly linear, it is fairly linear. Use the data from 1976 and 2010 to find an explicit formula for the number of stay-at-home fathers, then use it to predict the number of stay-at-home fathers in 2020.

\(\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1976 & 1984 & 1991 & 2000 & 2010 \\ \hline \text { Number of stay-at-home fathers } & 20,610 & 28,725 & 43,530 & 47,665 & 53,555 \\ \hline \end{array}\)

We let \(t\) = the number of years after 1976, and let \(y\) = the number of stay-at-home fathers.

From the table we know that 1976 corresponds to \(t=0\) and the number of stay-at-home fathers is \(y=20,610\).

From 1976 to 2010 the number of stay-at-home fathers increased by

\(53,555-20,610=32,945\)

This happened over 34 years, so the rate of change (slope) is \(32,945 / 34=969\).

\(y=969t+20,610\)

Predicting for 2020, we use \(t=44\)

\(y=969(44)+20,610=63,246\)

There will be 63,246 stay-at-home fathers in 2020.

[1] www.fira.ca/article.php?id=140

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  • 1. Multiple Choice 5 minutes 1 pt Which of the following situations could be descried by the equation y=120-25x? There are 120 people in the football stadium, and 25 more are entering each hour. A plumber charges $25 for a house call and $120 per hour. A teacher has 120 students. Her third period has 25 students There are 120 gallons of water in a tank. It releases water at a rate of 25 gallons per minute
  • 3. Multiple Choice 3 minutes 1 pt The equation below represents an elevator's height after an amount of seconds. Using this equation, what should the height of the elevator be after 14 seconds? h(s) = -3.8s + 290 80 236.8 -80 72.6
  • 4. Multiple Choice 2 minutes 1 pt The amount of water in the bucket ( in pints) can be modeled by the equation V(n) = 2n + 7 where n is the number of pitchers used to put water in the bucket.  If the bucket contains 29 pints of water, how many pitchers of water were used to fill it? 11 65 9 18
  • 5. Multiple Choice 2 minutes 1 pt The commission of a sales representative can be modeled by C(x) = 4n - 10, where n is the number of items sold.  If the sales representative sales 30 items, how much commission will they earn? $20 $120 $5 $110
  • 6. Multiple Choice 2 minutes 1 pt The GPA of a student can be predicted using the equation G(h) = 1.4h + 1.4  where h is the hours the student spends studying.  How many hours did a student spend studying if they have a GPA of 4.2? 4 3 not enough information 2
  • 7. Multiple Choice 2 minutes 1 pt The GPA of a student can be predicted using the equation G(h) = 1.4h + 1.4  where h is the hours the student spends studying.  What is the expected GPA of a student that spends no hours studying? 2.4 1.4 not enough information 2.8
  • 8. Multiple Choice 30 seconds 1 pt R(n)=125 + 80n models the amount of money R(n) in the register after selling n tickets. How much money is in the register before selling any tickets?  $125 $80 $205 $45
  • 9. Multiple Choice 30 seconds 1 pt R(n)=125 + 80n models the amount of money R(n) in the register after selling n tickets. How many tickets must be sold to have $18,445 in the register?  233 92 229 119
  • 10. Multiple Choice 1 minute 1 pt A driving range charges $4 to rent a golf club plus $2.75 for every bucket of golf balls you hit. Write an equation that shows the total cost c of hitting b buckets of golf balls. c = 2.75b + 4 c = 2.75b c =  4b- 2.75 c = 2.75b - 4
  • 11. Multiple Choice 5 minutes 1 pt A fitness club opens with 80 members. Each month the membership increases by 15 members. Which equation represents the relationship between the number of months the club has been opened, n, and the total fitness club membership, M? M(n)=15n M(n)=15n+80 M(n)=n+15 M(n)=80n+15
  • 12. Multiple Choice 15 minutes 1 pt The function defined by f(n) = 12n + 500 represents the total amount given to charity as a function of the number of students present at the event. If the school would like to donate $3000 to its favorite charity, how many students will have to attend to make that happen? 229 219 239 209
  • 13. Multiple Choice 1 minute 1 pt Asia is a salesperson for Quark Computer Company. Each month she earns $2500 as well as 1/6 of every sale she makes. What equation represents this situation? C(s) = 6s - 2500 C(s) = 6s + 2500 C(s) = (1/6)s - 2500 C(s) = (1/6)s + 2500
  • 14. Multiple Choice 1 minute 1 pt Asia is a salesperson for Quark Computer Company. Each month she earns $2500 as well as 1/6 of every sale she makes. How much did Asia make after 900 total sales? $2650 $3100 $2950 $7900
  • 15. Multiple Choice 1 minute 1 pt Asia is a salesperson for Quark Computer Company. Each month she earns $2500 as well as 1/6 of every sale she makes. How many sales did she make to earn $2700 last month? 1150 1000 1250 1200
  • 16. Multiple Choice 30 seconds 1 pt The value of a particular car model t years after its purchased is given by V ( t ) = 25000-3000 t What does V ( 0 ) represent? Value after 1 year Value after 1 week Value after 1 month Purchase price of the car
  • 17. Multiple Choice 30 seconds 1 pt The value of a particular car model t years after its purchased is given by V ( t ) = 25000-3000 t What is the cars value 6 years after it is purchased? $6850 $7000 $7500 $8250
  • 18. Multiple Choice 30 seconds 1 pt The cost of hiring a private tutor is modeled by the function C(n)=32n+20. How much does the tutor charge per hour? $32 per hour $20 per hour $52 per hour $12 per hour
  • 19. Multiple Choice 30 seconds 1 pt The cost of hiring a private tutor is modeled by the function C(n)=32n+ 20 . What does 20 represent in this model? Price per hour Flat fee to hire tutor Maximum number of hour you can hire the tutor I don't know what it means
  • 20. Multiple Choice 30 seconds 1 pt The cost of hiring a private tutor is modeled by the function C(n)=32n+20. How many hour did the tutor work to earn $212? 4 hours 6 hours 5 hours 7 hours

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Solve Systems of Linear Equations: Elimination

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"Solve Systems of Linear Equations: Elimination" is an i-Ready Math Level H lesson . The domain is Algebra and Algebraic Thinking, and the code is di.math.al.8.0530. It was introduced in October 2022.

  • This is the first lesson focusing on this concept.
  • Anyone who finished i-Ready (finished all the lessons) in math before October 2022 did not do any lesson focusing on this.
  • The concept taught in this lesson is high-school level as it is usually fully taught in Algebra I. However it can sometimes be taught just a little bit in eighth grade.

Objectives [ ]

• Use elimination to solve systems of linear equations.

• Identify efficient ways to solve a system of linear equations.

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  • 1 Solve Systems of Linear Equations: Elimination
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  1. Equations for Linear Models

    Trivia If you didn't do this lesson in the 2022-23 school year, you probably did its replacement due to this lesson not being introduced until October 2022. It is never cleared up what a Linear Model is. Instead they call it a "line of fit." This lesson builds on concepts taught in Analyze Scatter Plots and Fit a Linear Model. Gallery Categories

  2. Ready Mathematics Practice and Problem Solving Grade 8

    Textbook solutions Verified Chapter 1: Expressions and Equations (Exponents) and the Number System Section Lesson 1: Properties of Integer Exponents Section Lesson 2: Square Roots and Cube Roots Section Lesson 3: Understand Rational and Irrational Numbers Section Lesson 4: Scientific Notation Section Lesson 5: Performance Task Page 3:

  3. Solving linear equations "i-Ready"

    Mathematics 8th grade Solving linear equations "i-Ready" Angela Caicedo 108 plays 8 questions Copy & Edit Live Session Assign Show Answers See Preview 1. Multiple Choice 30 seconds 1 pt Solve the equation. 8 (4 - x) = 7x + 2 x = 5 x = -6 x = 2 x ≠ 0 2. Multiple Choice 30 seconds 1 pt What value of m makes the equation true? m + 2m - 6 = -12 + 2m

  4. Problem Solving with Linear Models

    Answer: extrapolate the data in a linear manner to predicate a value Explanation: For example making a graph of temperature vs volume can be used to predict a value for absolute zero. The linear line can be extended to find a predicted value for the temperature where there is zero volume.

  5. Solving Real World Problems with Systems of Linear Equation #1

    Solving real world problems with systems of linear equations Common Core State Standard (CCSS): 8EE8c - Solve real-world and mathematical problems leading to two linear equations in two...

  6. Chapter 2 Part 1: Linear Function Models and Problem Solving

    Chapter 2 Part 1: Linear Function Models and Problem Solving. Term. 1 / 18. Average Rate of Change. Click the card to flip 👆. Definition. 1 / 18. In Algebra terms it is the slope of the line; in the broader sense it is how fast one variable is changing compared to another variable. Click the card to flip 👆.

  7. 2.5.4: Applications Using Linear Models

    Applying Linear Models. Modeling linear relationships can help solve real-world applications. Consider the example situations below, and note how different problem-solving methods may be used in each. Nadia has $200 in her savings account. She gets a job that pays $7.50 per hour and she deposits all her earnings in her savings account.

  8. 4.2: Solving Linear Systems by Substitution

    Solve linear systems using the substitution method. The Substitution Method. In this section, we will define a completely algebraic technique for solving systems. The idea is to solve one equation for one of the variables and substitute the result into the other equation. ... Answer: \((9, 6)\). The check is left to the reader. Example ...

  9. 4.2: Modeling with Linear Functions

    The rate of change is constant, so we can start with the linear model M(t) = mt + b M ( t) = m t + b. Then we can substitute the intercept and slope provided. To find the x-intercept, we set the output to zero, and solve for the input. 0 t = −400t + 3500 = 3500 400 = 8.75 0 = − 400 t + 3500 t = 3500 400 = 8.75.

  10. 7.2: Modeling with Linear Equations

    By substituting one of the points in the equation, we get b = 500 b = 500. Therefore, the cost equation is y = 10x + 500 y = 10 x + 500. This equation is the linear model for this problem. Now use the linear model to find the cost of 100 items. Substitute x = 100 x = 100 in the equation y = 10x + 500 y = 10 x + 500. So the cost is.

  11. Linear Functions: Model From Two Points Level H I-ready Lesson Answers

    98 Share 15K views 1 year ago Answers to the Linear Functions: Model From Two Points Level H I-ready Lesson ...more ...more Math Task #131 - Linear Function: Rate of Change and Initial...

  12. Linear equation word problems

    This means that we have to multiply the number of classes that she will take: c by 2 1/2 to find the total number of hours she will spend doing homework total. The equation that we have now is: 2 1/2c + ... = 19. - Finally, we are told that Kaylee will spend an additional 6 1/2 hours total doing assigned readings, not 6 1/2 hours for each class.

  13. Linear models word problem: book (video)

    A linear function has a correlation coefficient of 1 (or -1) while a linear model can have a correlation coefficient less than one, the closer it is to 1, the closer that the scatterplot is to the linear model of an equation. Thus, a linear function is a specific type of linear models, but there are many more linear models that are not linear ...

  14. Linear models word problems (practice)

    Lesson 9: Linear models. Linear graphs word problems. Modeling with tables, equations, and graphs. Linear graphs word problem: cats. Linear equations word problems: volcano. Linear equations word problems: earnings. Modeling with linear equations: snow. Linear equations word problems: graphs. Linear equations word problems.

  15. Problem Solving with Linear Models

    Using Dana's data from the first problem, estimate the circumference of a circle whose diameter is 60 inches. The equation y = 3.14x + 0.42 of the relationship between diameter and circumference from the first problem applies here. Diameter = 60 inches ⇒ y = 3.14(60) + 0.42 = 188.82 inches _. A circle with a diameter of 60 inches will have a ...

  16. Ready Mathematics: Practice and Problem Solving Grade 7

    Curriculum Associates Textbook solutions Verified Chapter Unit 1: The Number System Section 1: Understand Addition of Positive and Negative Integers Section 2: Understand Subtracting of Positive and Negative Integers Section 3: Add and Subtract Positive and Negative Integers Section 4: Multiple and Divide Positive and Negative Integers Section 5:

  17. Graph Systems of Linear Equations

    The code is di.math.al.8.0510. It was introduced in July 2022. Solve systems of linear equations by graphing. Despite its only objective, this lesson has a lot more to it. This lesson is the first lesson on its...

  18. PDF LESSON Lesson 29 OVERVIEW Graph Points in the Coordinate Plane

    Problem Solving Assign pages 315-316. Day 3 45-60 minutes Modeled and Guided Instruction Learn About Solving Problems on a Coordinate Plane • Picture It/Model It 15 min • Connect It 20 min • Try It 10 min Practice and Problem Solving Assign pages 317-318. Day 4 45-60 minutes Guided Practice Practice Solving Problems on a ...

  19. Linear equations, functions, & graphs

    Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities.

  20. Linear Models

    Linear Models quiz for 9th grade students. Find other quizzes for Mathematics and more on Quizizz for free! ... Linear Equations 6.9K plays 8th 19 Qs . Solving Linear Equations 627 plays 7th - 8th 10 Qs . Unit Rate 2.3K plays 5th - 6th 20 Qs . Linear Equations Word Problems 2K plays 8th - 12th Browse from millions of quizzes. QUIZ . Linear ...

  21. Solve Systems of Linear Equations: Elimination

    The thumbnail Objective Eliminate one variable to solve a System of Linear Equations. Characters i-Ready Ball (intro only) Level Level H: High-Level Lesson Chronology Previous Lesson Solve Systems of Linear Equations: Substitution Next Lesson Understand Functions Trivia This is the first lesson focusing on this concept.

  22. Linear Functions: Model From a Verbal Description Level H I ...

    Answers to the Linear Functions: Model From a Verbal Description Level H I-Ready Lesson

  23. Mathway

    Free math problem solver answers your algebra homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. ... Mathway currently only computes linear regressions. We are here to assist you with your math questions. You will need to get assistance from your school if you are having ...