8.EE.D: Analyze and solve linear equations and pairs of...

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by… (more)



Sort the following equations into groups with either no solution, one solution or infinitely many solutions: \(x = \frac{2}{5}\), \(3h = 12\), \(4(n - 2) - 1 = 5n - n + 9\), \(r = 5r -3r - r\), \(2j + 7 = 2j + 7\), \(\frac{2}{10} = 0.02\), \(4 = 9\)
Sort the following equations into groups with either no solution, one solution or infinitely many solutions: \(x = \frac{2}{5}\), \(3h = 12\), \(4(n - 2) - 1 = 5n - n + 9\), \(r = 5r -3r - r\), \(2j + 7 = 2j + 7\), \(\frac{2}{10} = 0.02\), \(4 = 9\)
Identify equations with 0, 1, infinite solutions

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and… (more)



Temperature is measured in both Celsius and Fahrenheit degrees. These two systems are of course related: the \(Fahrenheit\) temperature is obtained by adding 32 to 9/5 of the \(Celsius\) temperature. In the following questions, let \(C\) represent the Celsius temperature and \(F\) the Fahrenheit temperature.

(a) Write an equation that expresses \(F\) in terms of \(C\).

(b) Use this equation to find the value of \(F\) that corresponds to \(C=20\).

(c) On the Celsius scale, water freezes at \(0^{\mathrm{o}}\) and boils at \(100^{\mathrm{o}}\) Use your formula to find the corresponding temperatures on the Fahrenheit scale. Do you recognize your answers?

(d) A quick way to get an approximate Fahrenheit temperature from a Celsius temperature is to double the Celsius temperature and add 30. Explain why this is a good approximation. Convert \(23^{\mathrm{o}}\) Celsius the quick way. What is the difference between your answer and the correct value? For what Celsius temperature does the quick way give the correct value?
Temperature is measured in both Celsius and Fahrenheit degrees. These two systems are of course related: the \(Fahrenheit\) temperature is obtained by adding 32 to 9/5 of the \(Celsius\) temperature. In the following questions, let \(C\) represent the Celsius temperature and \(F\) the Fahrenheit temperature.

(a) Write an equation that expresses \(F\) in terms of \(C\).

(b) Use this equation to find the value of \(F\) that corresponds to \(C=20\).

(c) On the Celsius scale, water freezes at \(0^{\mathrm{o}}\) and boils at \(100^{\mathrm{o}}\) Use your formula to find the corresponding temperatures on the Fahrenheit scale. Do you recognize your answers?

(d) A quick way to get an approximate Fahrenheit temperature from a Celsius temperature is to double the Celsius temperature and add 30. Explain why this is a good approximation. Convert \(23^{\mathrm{o}}\) Celsius the quick way. What is the difference between your answer and the correct value? For what Celsius temperature does the quick way give the correct value?
Phillips Exeter Academy (no answers available)  ■  edit


Because \(12x^{2}+5x^{2}\) is equivalent to \(17x^{2}\), the expressions \(12x^{2}\) and \(5x^{2}\) are called \({\it like~terms}\). Explain. Why are \(12x^{2}\) and \(5x\) called \({\it unlike~terms}\)? Are \(3ab\) and 11ab like terms? Explain. Are \(12x^{2}\) and \(5y^{2}\) like terms? Explain. Are \(12x^{2}\) and \(12x\) like terms? Explain.
Because \(12x^{2}+5x^{2}\) is equivalent to \(17x^{2}\), the expressions \(12x^{2}\) and \(5x^{2}\) are called \({\it like~terms}\). Explain. Why are \(12x^{2}\) and \(5x\) called \({\it unlike~terms}\)? Are \(3ab\) and 11ab like terms? Explain. Are \(12x^{2}\) and \(5y^{2}\) like terms? Explain. Are \(12x^{2}\) and \(12x\) like terms? Explain.
Phillips Exeter Academy (no answers available)  ■  edit


Solve the following for \(x\):

(a) \(4-(x+3)=8-5(2x-3)\)
(b) \(x-2(3-x)=2x+3(1-x)\)
Solve the following for \(x\):

(a) \(4-(x+3)=8-5(2x-3)\)
(b) \(x-2(3-x)=2x+3(1-x)\)
Phillips Exeter Academy (no answers available)  ■  edit
Solve linear equations: multi-step

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection… (more)



(a) By completing the table of values, draw the lines y = 2x + 3 and x = 1 - 2y on the grid.

(b) Do the equations y = 2x + 3 and x = 1 - 2y have one common solution, no common solutions, or infinitely many common solutions? Explain how you know.
(c) Draw a straight line on the grid that has no common solutions with the line y = 2x + 3. What is the equation of your new line? Explain your answer.
(a) By completing the table of values, draw the lines y = 2x + 3 and x = 1 - 2y on the grid.

(b) Do the equations y = 2x + 3 and x = 1 - 2y have one common solution, no common solutions, or infinitely many common solutions? Explain how you know.
(c) Draw a straight line on the grid that has no common solutions with the line y = 2x + 3. What is the equation of your new line? Explain your answer.



(a) By completing the table of values, draw the lines y = 2x + 2 and x = 4 - 2y on the grid.

(b) Do the equations y = 2x + 2 and x = 4 - 2y have one common solution, no common solutions, or infinitely many common solutions? Explain how you know.
(c) Draw a straight line on the grid that has no common solutions with the line y = 2x + 2. What is the equation of your new line? Explain your answer.

(a) By completing the table of values, draw the lines y = 2x + 2 and x = 4 - 2y on the grid.

(b) Do the equations y = 2x + 2 and x = 4 - 2y have one common solution, no common solutions, or infinitely many common solutions? Explain how you know.
(c) Draw a straight line on the grid that has no common solutions with the line y = 2x + 2. What is the equation of your new line? Explain your answer.


Randy and Sandy have a total of 20 books between them. After Sandy loses three by leaving them on the bus, and some birthday gifts double Randy's collection, their total increases to 30 books. How many books did each have before these changes?
Randy and Sandy have a total of 20 books between them. After Sandy loses three by leaving them on the bus, and some birthday gifts double Randy's collection, their total increases to 30 books. How many books did each have before these changes?
Phillips Exeter Academy (no answers available)  ■  edit


Does every system of equations \(px+qy=r\) and \(mx+ny =k\) have a simultaneous solution \((x, y)\) ? Explain.
Does every system of equations \(px+qy=r\) and \(mx+ny =k\) have a simultaneous solution \((x, y)\) ? Explain.
Phillips Exeter Academy (no answers available)  ■  edit
Identify systems with 0, 1, infinite solutions

Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine… (more)



Lindsay Lohan and Macklemore are about to record an album together. They have a choice between two recording studios. One studio charges a $50 flat fee plus $2 per minute. The second charges a $20 flat fee plus $4 per minute. Lindsay Lohan thinks they should go with the second studio. Macklemore thinks they should go with the first. For what numbers of minutes would each studio make sense? Use equations and a graph to justify your response.
Lindsay Lohan and Macklemore are about to record an album together. They have a choice between two recording studios. One studio charges a $50 flat fee plus $2 per minute. The second charges a $20 flat fee plus $4 per minute. Lindsay Lohan thinks they should go with the second studio. Macklemore thinks they should go with the first. For what numbers of minutes would each studio make sense? Use equations and a graph to justify your response.
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Kimi and Jordan are both working during the summer to earn money in addition to their weekly allowances, and they are saving all their money. Kimi earns $9 an hour at her job, and her allowance is $8 per week. Jordan earns $7.50 an hour, and his allowance is $16 per week.

(a) Complete the table below:

(b) Write an equation that can be used to calculate the total of Kimi's allowance and job earnings at the end of one week given the number of hours she works.
(c) Write an equation that can be used to calculate the total of Jordan's allowance and job earnings at the end of one week given the number of hours he works.
(d) Sketch the graphs of your two equations on a pair of axes.
(e) Jordan wonders who will save more money in one week if they both work the same number of hours. Write an answer for him.
Kimi and Jordan are both working during the summer to earn money in addition to their weekly allowances, and they are saving all their money. Kimi earns $9 an hour at her job, and her allowance is $8 per week. Jordan earns $7.50 an hour, and his allowance is $16 per week.

(a) Complete the table below:

(b) Write an equation that can be used to calculate the total of Kimi's allowance and job earnings at the end of one week given the number of hours she works.
(c) Write an equation that can be used to calculate the total of Jordan's allowance and job earnings at the end of one week given the number of hours he works.
(d) Sketch the graphs of your two equations on a pair of axes.
(e) Jordan wonders who will save more money in one week if they both work the same number of hours. Write an answer for him.


Cee Lo Green and Rihanna are both on Twitter. Cee Lo Green has 125,000 followers with 500 joining everyday. Rihanna has 97,000 followers with 1,200 joining everyday. How long until Rihanna has more Twitter followers than Cee Lo Green? Write equations, make a graph, and solve the system to provide a precise answer.
Cee Lo Green and Rihanna are both on Twitter. Cee Lo Green has 125,000 followers with 500 joining everyday. Rihanna has 97,000 followers with 1,200 joining everyday. How long until Rihanna has more Twitter followers than Cee Lo Green? Write equations, make a graph, and solve the system to provide a precise answer.
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You work for a video streaming company that has two monthly plans to choose from.
Plan 1: A flat rate of $7 per month plus $2.50 per video viewed.
Plan 2: $4 per video viewed.
(a) What type of functions model this situation? Explain how you know.
(b) Define variables that make sense in the context, and then write an equation for each plan with cost as a function of videos viewed.
(c) How much would 3 videos in a month cost for each plan? 5 videos?
(d) Compare the two plans and explain what advice you would give to a customer trying to decide which plan is best for them, based on their viewing habits.
You work for a video streaming company that has two monthly plans to choose from.
Plan 1: A flat rate of $7 per month plus $2.50 per video viewed.
Plan 2: $4 per video viewed.
(a) What type of functions model this situation? Explain how you know.
(b) Define variables that make sense in the context, and then write an equation for each plan with cost as a function of videos viewed.
(c) How much would 3 videos in a month cost for each plan? 5 videos?
(d) Compare the two plans and explain what advice you would give to a customer trying to decide which plan is best for them, based on their viewing habits.
—Illustrative Mathematics  ■  edit
Systems of linear equations in context



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