Algebra Quiz: Creating And Graphing Two Variable Equations
20 questions · exam conditions
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Creating And Graphing Two Variable EquationsQuestion 1 of 20

A movie theater charges a $4 booking fee plus $9 for each ticket. Let $Cbethetotalcost(indollars)andletbe the total cost (in dollars) and lettbethenumberoftickets.Whatequationrepresentstherelationshipbetweenbe the number of tickets. What equation represents the relationship betweentandandC$?

C=4tC = 4t
C=13tC = 13t
C=9+4tC = 9 + 4t
C=9t+4C = 9t + 4
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Algebra Quiz

Algebra Quiz: Creating And Graphing Two Variable Equations

Practice Creating And Graphing Two Variable Equations in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Creating And Graphing Two Variable Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A movie theater charges a $4 booking fee plus $9 for each ticket. Let $Cbethetotalcost(indollars)andletbe the total cost (in dollars) and lettbethenumberoftickets.Whatequationrepresentstherelationshipbetweenbe the number of tickets. What equation represents the relationship betweentandandC$?

  1. C=4tC = 4t
  2. C=13tC = 13t
  3. C=9+4tC = 9 + 4t
  4. C=9t+4C = 9t + 4 (correct answer)
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, the theater charges $9 per ticket plus a $4 booking fee, we identify that total cost C depends on number of tickets t. The rate is $9 per ticket (that becomes our coefficient), and the booking fee is $4 (that's our constant term). So the equation is C = 9t + 4. This equation lets us calculate the total cost for any number of tickets! Choice C is correct because it accurately represents the relationship with $9 per ticket (9t) plus the $4 booking fee (+4). Choice A has the numbers switched: it puts $4 per ticket and a $9 fee, but the context tells us it's $9 per ticket and a $4 fee. When translating words to equations, make sure each part of the equation corresponds to something in the description! Quick trick: the words in the problem often tell you what operation to use. 'Per' or 'each' usually means multiply (like $9 per ticket = 9 times number of tickets). 'Plus' or 'and' means add (like $4 fee plus ticket cost). Listen to the language!

Question 2

A streaming service charges a $5 sign-up fee and then $8 per month. If $misthenumberofmonthsandis the number of months andC$ is the total cost (in dollars), what should the axes be labeled when graphing this relationship on a coordinate plane?

  1. x-axis: Total Cost ($); y-axis: Months
  2. x-axis: Months; y-axis: Total Cost ($) (correct answer)
  3. x-axis: Months ($); y-axis: Total Cost (months)
  4. x-axis: xx; y-axis: yy
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. The coordinate plane helps us visualize relationships: the horizontal x-axis typically shows the independent variable (the one you choose or that changes first, like time or quantity), while the vertical y-axis shows the dependent variable (the one that responds, like cost or height). For this relationship, the x-axis should represent months with label 'Months', and the y-axis should represent total cost with label 'Total Cost ()′,whichmakessensebecausemonthsistheindependentvariable(youchoosehowmanymonths)andcostdependsonthatchoice.ChoiceBiscorrectbecauseitsetsuptheaxesappropriatelywithmonthsonxandcostony,includingthedollarunitsforcost.ChoiceAhasthevariablesswitched:itputstotalcostonthex−axisandmonthsonthey−axis,butremember—theindependentvariable(theoneyoustartwithorcontrol)goesonthex−axis,andthedependentvariable(theonethatresponds)goesonthey−axis.Rememberthedifferencebetweenindependentanddependentvariables:theindependentvariableistheoneyoucanchooseorcontrol(likehowmanymonthsyousubscribe),andthedependentvariableistheonethatrespondstoyourchoice(likewhatthetotalcostis)—independentgoesonthex−axis,dependentonthey−axis—thisisthestandardconvention!Forgraphing,thinkaboutyouraudience:goodaxislabelsincludethevariablenameANDunits(like′TotalCost()', which makes sense because months is the independent variable (you choose how many months) and cost depends on that choice. Choice B is correct because it sets up the axes appropriately with months on x and cost on y, including the dollar units for cost. Choice A has the variables switched: it puts total cost on the x-axis and months on the y-axis, but remember—the independent variable (the one you start with or control) goes on the x-axis, and the dependent variable (the one that responds) goes on the y-axis. Remember the difference between independent and dependent variables: the independent variable is the one you can choose or control (like how many months you subscribe), and the dependent variable is the one that responds to your choice (like what the total cost is)—independent goes on the x-axis, dependent on the y-axis—this is the standard convention! For graphing, think about your audience: good axis labels include the variable name AND units (like 'Total Cost ()' not just 'C'), and a good scale shows your data without bunching it up or spreading it too thin.

Question 3

A school fundraiser sells bracelets for $5 each and charges a one-time $10 setup fee for the order. Let $bbethenumberofbraceletsandletbe the number of bracelets and letCbethetotalcost(indollars).Whatequationrepresentstherelationshipbetweenbe the total cost (in dollars). What equation represents the relationship betweenCandandb$?​

  1. C=5bC = 5b
  2. C=10−5bC = 10 - 5b
  3. C=10b+5C = 10b + 5
  4. C=5b+10C = 5b + 10 (correct answer)
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, bracelets cost $5 each plus a one-time $10 setup fee, so the total cost C depends on the number of bracelets b. The rate is $5 per bracelet (that becomes our coefficient), and the starting amount is $10 (that's our constant term). So the equation is C = 5b + 10. This equation lets us calculate the total cost for any number of bracelets! Choice C is correct because it accurately represents the relationship with $5 per bracelet (5b) plus the $10 setup fee. Choice A has the numbers switched: it puts $10 per bracelet and a $5 fee, but the problem clearly states $5 is per bracelet and $10 is the setup fee. When translating words to equations, make sure each part of the equation corresponds to something in the description! Quick trick: the words in the problem often tell you what operation to use. 'Per' or 'each' usually means multiply (like $5 per bracelet = 5 times number of bracelets). 'Plus' or 'and' means add (like $10 fee plus bracelet cost). Listen to the language!

Question 4

A streaming service charges $10 per month plus a one-time setup fee of $5. Let $mbemonthsandbe months andC$ be total cost (in dollars). What is an appropriate scale to graph this relationship if you want to show from 0 to 6 months?

  1. x-axis: 0 to 6 by 1; y-axis: 0 to 70 by 10 (correct answer)
  2. x-axis: 0 to 60 by 10; y-axis: 0 to 6 by 1
  3. x-axis: 0 to 6 by 0.1; y-axis: 0 to 700 by 100
  4. x-axis: 0 to 6 by 2; y-axis: 0 to 20 by 1
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. Axis labels should be specific and include units: instead of just 'x' and 'y', write 'Time (hours)' and 'Distance (miles)' so anyone looking at your graph immediately understands what the numbers represent. Looking at the context, months range from 0 to 6, so a good scale for the x-axis would be marking every 1 unit. The cost starts at $5 (setup fee) and after 6 months reaches $5 + $10(6) = $65, so the y-axis should go from 0 to 70, marking every 10 units works well. This scale shows the data clearly without cramming too much or spreading it too thin! Choice A is correct because it chooses a reasonable scale with x-axis from 0 to 6 by 1 (perfect for months) and y-axis from 0 to 70 by 10 (captures the cost range nicely). Choice D uses a scale that's not practical: with costs ranging up to $65, having the y-axis only go to 20 would cut off most of the graph. A better scale shows all the data points clearly. For graphing, think about your audience: good axis labels include the variable name AND units (like 'Time (hours)' not just 't'), and a good scale shows your data without bunching it up or spreading it too thin. If your values go from 0 to 50, try marking every 5 or 10—not every 1 (too crowded) or every 100 (too sparse).

Question 5

The relationship between two numbers is described as: "yy is 7 less than three times xx." What equation represents this relationship?

  1. y=3x+7y = 3x + 7
  2. y=7x−3y = 7x - 3
  3. y=3x−7y = 3x - 7 (correct answer)
  4. y=3(x−7)y = 3(x - 7)
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like y and x), choose variables to represent them, then write an equation that captures how one depends on the other. From the context, y is 7 less than three times x, we identify that y depends on x; the rate is 3 (that becomes our coefficient), and we subtract 7 (that's our constant term), so the equation is y = 3x - 7, which lets us calculate y for any x! Choice C is correct because it accurately represents the relationship with the multiplication by 3 and then subtracting 7. Choice A has the math operation wrong: the context says '7 less than,' which means subtract 7, but this choice adds 7—when you see words like 'less than,' that usually means subtraction! Quick trick: the words in the problem often tell you what operation to use—'times' means multiply (like three times x = 3x), 'less than' means subtract (like 7 less = -7). When creating equations from word problems, ask yourself three questions: (1) What are the two quantities that are related? (2) Which one changes independently (that's your x), and which one depends on the first (that's your y)? (3) What's the mathematical relationship—constant rate (linear), area/product (quadratic), or something else? Answer these, and writing the equation becomes much easier!

Question 6

A streaming service charges $9 per month with no extra fees. Let $mbethenumberofmonthsandbe the number of months andCbethetotalcostindollars.Writeanequationtorepresenttherelationshipbetweenbe the total cost in dollars. Write an equation to represent the relationship betweenmandandC$ for graphing.

  1. C=9mC = 9m (correct answer)
  2. C=9+mC = 9 + m
  3. m=9Cm = 9C
  4. C=m−9C = m - 9
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and months), choose variables to represent them (like C for cost and m for months), then write an equation that captures how one depends on the other. From the context, cost is $9 per month with no extra fees, we identify that C depends on m; the rate is 9 (that becomes our coefficient), and there's no starting amount (so constant is 0), so the equation is $C = 9m,whichletsuscalculatecostforanynumberofmonths!ChoiceAiscorrectbecauseitaccuratelyrepresentstherelationshipwiththemonthlyrateasthecoefficientandnoconstantterm.ChoiceBaddsaconstantincorrectly:ituses, which lets us calculate cost for any number of months! Choice A is correct because it accurately represents the relationship with the monthly rate as the coefficient and no constant term. Choice B adds a constant incorrectly: it uses C = 9 + m,butthecontexthasnofixedfee,just$9permonth;whentranslatingwordstoequations,makesurenottoaddextrapartsthataren′tdescribed!Rememberthedifferencebetweenindependentanddependentvariables:theindependentvariableistheoneyoucanchooseorcontrol(likenumberofmonths),andthedependentvariableistheonethatresponds(liketotalcost);independentgoesonthex−axis,dependentonthey−axis—thisisthestandardconvention!Afteryoucreateyourequation,testitwithsimplevalues:try$m=1, but the context has no fixed fee, just $9 per month; when translating words to equations, make sure not to add extra parts that aren't described! Remember the difference between independent and dependent variables: the independent variable is the one you can choose or control (like number of months), and the dependent variable is the one that responds (like total cost); independent goes on the x-axis, dependent on the y-axis—this is the standard convention! After you create your equation, test it with simple values: try $m = 1 (should give $9) and $m = 2$ (should give $18); if it works, you're set!

Question 7

A movie theater charges $8 for a ticket plus $3 for each snack. Let $Cbethetotalcost(indollars)andletbe the total cost (in dollars) and letsbethenumberofsnacks.Whatequationrepresentstherelationshipbetweenbe the number of snacks. What equation represents the relationship betweenCandands$?

  1. C=8s+3C = 8s + 3
  2. C=8+3sC = 8 + 3s (correct answer)
  3. C=11sC = 11s
  4. C=3+8sC = 3 + 8s
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, the theater charges $8 for a ticket (a fixed cost) plus $3 for each snack (a variable cost), we identify that total cost C depends on number of snacks s. The fixed ticket cost is $8 (that's our constant term), and the rate per snack is $3 (that becomes our coefficient for s). So the equation is C = 8 + 3s. This equation lets us calculate the total cost for any number of snacks! Choice B is correct because it accurately represents the relationship with $8 as the fixed ticket cost and $3s as the cost that varies with the number of snacks. Choice A has the same numbers but switches their roles: it makes the snack cost $8 each and the ticket only $3, but remember—the problem says the ticket is $8 and each snack is $3. When translating words to equations, make sure each part of the equation corresponds to something in the description! Quick trick: the words in the problem often tell you what operation to use. 'Per' or 'each' usually means multiply (like $3 per snack = 3 times number of snacks). 'Plus' or 'and' means add (like $8 ticket plus snack cost). Listen to the language!

Question 8

A movie theater charges a $6 admission fee plus $2 for each snack a student buys. Let $nbethenumberofsnacksandbe the number of snacks andC$ be the total cost in dollars. What equation represents this relationship so it can be graphed on a coordinate plane?

  1. C=6n+2C = 6n + 2
  2. n=2C+6n = 2C + 6
  3. C=8nC = 8n
  4. C=2n+6C = 2n + 6 (correct answer)
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of snacks), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, cost is $6 admission plus $2 per snack, we identify that C depends on n; the rate is 2 (that becomes our coefficient), and the starting amount is 6 (that's our constant term), so the equation is C = 2n + 6, which lets us calculate cost for any number of snacks! Choice B is correct because it accurately represents the relationship with the fixed fee as the constant and the per-snack cost as the coefficient. Choice D has the coefficients switched: it uses 6n + 2, but the context tells us $2 per snack (not $6) and $6 fixed (not $2); when translating words to equations, make sure each part corresponds to the description! When creating equations from word problems, ask yourself three questions: (1) What are the two quantities that are related? (2) Which one changes independently (that's your x), and which one depends on the first (that's your y)? (3) What's the mathematical relationship—constant rate (linear), area/product (quadratic), or something else? Answer these, and writing the equation becomes much easier! After you create your equation, test it with simple values: if C = 2n + 6, try n = 0 (should give $6) and n = 1 (should give $8); if it matches, you're good!

Question 9

A relationship is described by: "yy is 4 less than three times xx." What equation represents this relationship?​

  1. y=3(x−4)y = 3(x - 4)
  2. y=3x+4y = 3x + 4
  3. y=3x−4y = 3x - 4 (correct answer)
  4. y=4x−3y = 4x - 3
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, 'y is 4 less than three times x' means we first multiply x by 3 (giving 3x), then subtract 4. So y = 3x - 4. This equation lets us calculate y for any value of x! Choice C is correct because it accurately represents the relationship: three times x (3x) minus 4, which matches '4 less than three times x'. Choice A has the wrong operation: it adds 4 instead of subtracting, giving us '4 more than three times x' instead of '4 less than'. When you see 'less than', that means subtract! Quick trick: the words in the problem often tell you what operation to use. 'Times' means multiply (three times x = 3x). 'Less than' means subtract (4 less than something means something - 4). 'More than' means add (5 more than twice x = 2x + 5). Listen to the language!

Question 10

A relationship is described by: "yy is 5 less than three times xx." What equation represents this relationship?

  1. y=3x+5y = 3x + 5
  2. y=5x−3y = 5x - 3
  3. y=3x−5y = 3x - 5 (correct answer)
  4. y=3(x−5)y = 3(x-5)
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, 'y is 5 less than three times x', we need to translate each part: 'three times x' means 3x, and '5 less than' means subtract 5. So y equals 3x minus 5, giving us y = 3x - 5. This equation lets us calculate y for any value of x! Choice C is correct because it accurately represents the relationship y = 3x - 5 (three times x, then subtract 5). Choice A has the wrong operation: it adds 5 instead of subtracting, giving y = 3x + 5, but '5 less than' means subtract 5, not add 5. When you see 'less than,' that's your signal to subtract! Quick trick: the words in the problem often tell you what operation to use. 'Times' means multiply (three times x = 3x). 'Less than' means subtract (5 less than = -5). 'More than' would mean add. Listen to the language and translate piece by piece!

Question 11

A taxi charges a base fee of $4 plus $2.50 per mile. Let $Cbethetotalcost(indollars)andletbe the total cost (in dollars) and letmbethenumberofmilesridden.Whatequationrepresentstherelationshipbetweenbe the number of miles ridden. What equation represents the relationship betweenCandandm$?

  1. C=4m+2.5C = 4m + 2.5
  2. C=2.5m+4C = 2.5m + 4 (correct answer)
  3. C=2.5+4mC = 2.5 + 4m
  4. C=6.5mC = 6.5m
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and miles), choose variables to represent them (like C for cost and m for miles), then write an equation that captures how one depends on the other. From the context, cost is $4 base plus $2.50 per mile, we identify that C depends on m; the rate is 2.5 (that becomes our coefficient), and the starting amount is 4 (that's our constant term), so the equation is C = 2.5m + 4, which lets us calculate cost for any miles! Choice B is correct because it accurately represents the relationship with the base fee as the constant and the per-mile rate as the coefficient. Choice A has the numbers switched: it puts 4 as the coefficient and 2.5 as the constant, but the context tells us the base is $4 and per mile is $2.50—when translating words to equations, make sure each part corresponds to the description! Quick trick: the words in the problem often tell you what operation to use—'per' or 'each' usually means multiply (like $2.50 per mile = 2.5 times miles), 'plus' means add (like plus $4 base). After you create your equation, test it with simple values: if C = 2.5m + 4, try m = 0 (should give $4) and m = 1 (should give $6.50)—if it matches, great job!

Question 12

A taxi charges a $4 starting fee plus $2 per mile. Let $mbemilestraveledandbe miles traveled andC$ be total cost (in dollars). Which equation represents this relationship?

  1. C=6mC = 6m
  2. C=4m+2C = 4m + 2
  3. C=2m+4C = 2m + 4 (correct answer)
  4. m=2C+4m = 2C + 4
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, the taxi charges $4 to start plus $2 per mile, we identify that total cost C depends on miles traveled m. The rate is $2 per mile (that becomes our coefficient), and the starting amount is $4 (that's our constant term). So the equation is C = 2m + 4. This equation lets us calculate the total cost for any number of miles! Choice C is correct because it accurately represents the relationship with $2m as the per-mile charge and $4 as the starting fee. Choice B has the numbers switched: it makes the per-mile charge $4 and the starting fee only $2, but remember—the problem says the starting fee is $4 and each mile is $2. When translating words to equations, make sure each part of the equation corresponds to something in the description! Quick trick: the words in the problem often tell you what operation to use. 'Per' or 'each' usually means multiply (like $2 per mile = 2 times number of miles). 'Plus' or 'and' means add (like $4 starting fee plus mileage cost). Listen to the language!

Question 13

A gym charges a one-time sign-up fee of $25 and then $15 per month. Let $mbethenumberofmonthsandletbe the number of months and letC$ be the total cost (in dollars). Which equation models the total cost?

  1. C=25m+15C = 25m + 15
  2. C=15m+25C = 15m + 25 (correct answer)
  3. C=40mC = 40m
  4. C=15+25C = 15 + 25
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, the gym charges $25 once plus $15 per month, we identify that total cost C depends on number of months m. The rate is $15 per month (that becomes our coefficient), and the starting amount is $25 (that's our constant term). So the equation is C = 15m + 25. This equation lets us calculate the total cost for any number of months! Choice B is correct because it accurately represents the relationship with $15m as the monthly charges and $25 as the one-time sign-up fee. Choice A has the same numbers but switches their roles: it makes the monthly fee $25 and the sign-up fee only $15, but remember—the problem says the sign-up fee is $25 and each month is $15. When translating words to equations, make sure each part of the equation corresponds to something in the description! When creating equations from word problems, ask yourself three questions: (1) What are the two quantities that are related? (2) Which one changes independently (that's your x), and which one depends on the first (that's your y)? (3) What's the mathematical relationship—constant rate (linear), area/product (quadratic), or something else? Answer these, and writing the equation becomes much easier!

Question 14

A movie theater charges $9 for a ticket plus $3 for each snack you buy. Let $Cbethetotalcost(indollars)andletbe the total cost (in dollars) and lets$ be the number of snacks. What equation represents this relationship?

  1. C=12sC = 12s
  2. C=3s+9C = 3s + 9 (correct answer)
  3. C=9s+3C = 9s + 3
  4. C=9+3C = 9 + 3
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like cost and number of snacks), choose variables to represent them (like CC for cost and ss for snacks), then write an equation that captures how one depends on the other. From the context, the total cost is $9 for the ticket plus $3 per snack, we identify that $Cdependsondepends ons;therateis$3(thatbecomesourcoefficient),andthestartingamountis$9(that′sourconstantterm),sotheequationis$C=3s+9; the rate is $3 (that becomes our coefficient), and the starting amount is $9 (that's our constant term), so the equation is $C = 3s + 9, which lets us calculate the cost for any number of snacks! Choice C is correct because it accurately represents the relationship with the fixed ticket price as the constant and the snack cost as the variable term. Choice B has the numbers switched: it puts 9 as the coefficient and 3 as the constant, but the context tells us the ticket is the fixed $9 and snacks are $3 each—when translating words to equations, make sure each part corresponds to the description! When creating equations from word problems, ask yourself three questions: (1) What are the two quantities that are related? (2) Which one changes independently (that's your independent variable), and which one depends on the first (that's your dependent variable)? (3) What's the mathematical relationship—constant rate (linear), area/product (quadratic), or something else? Answer these, and writing the equation becomes much easier! Quick trick: the words in the problem often tell you what operation to use—'plus' or 'and' means add (like $9 plus $3 per snack = 9+3s9 + 3s).

Question 15

Two friends start a lawn care business. Their weekly revenue RR is R=25n+40R = 25n + 40, where nn is the number of lawns mowed. If they want to graph this relationship and show when they'll earn at least $200 per week, what should they include on their graph?

  1. Plot the line R=25n+40R = 25n + 40 and shade the region above the horizontal line R=200R = 200
  2. Plot the line R=25n+40R = 25n + 40 and mark the point where it intersects R=200R = 200 at (6.4,200)(6.4, 200)
  3. Plot the line R=25n+40R = 25n + 40 and shade the region to the right of the vertical line n=6.4n = 6.4
  4. Plot the line R=25n+40R = 25n + 40 and mark the point (6.4,200)(6.4, 200) with a note that they need at least 7 lawns (correct answer)
Explanation: To find when R≥200R ≥ 200: 200=25n+40200 = 25n + 40, so 160=25n160 = 25n, giving n=6.4n = 6.4. Since they can't mow partial lawns, they need at least 7 lawns. The graph should show the intersection point and note the practical constraint. Choice A shows revenue regions but not the constraint on nn, choice B doesn't address the practical interpretation, and choice C shades incorrectly.

Question 16

A bakery's daily profit PP depends on the number of specialty cakes cc sold according to P=15c−45P = 15c - 45. When graphing this relationship, what do the intercepts represent in this business context?

  1. yy-intercept: $45 loss when no cakes sold; $xx $-intercept: 3 cakes needed to break even (correct answer)
  2. yy-intercept: $45 profit when no cakes sold; $xx $-intercept: 3 cakes needed to break even
  3. yy-intercept: $45 loss when no cakes sold; $xx $-intercept: selling 3 cakes results in $0 profit
  4. yy-intercept: $15 profit per cake; $xx $-intercept: $45 in fixed costs to overcome
Explanation: When analyzing linear equations in real-world contexts, always examine what happens at the intercepts—they reveal crucial information about the situation when one variable equals zero. For P=15c−45P = 15c - 45, let's find both intercepts. The yy-intercept occurs when c=0c = 0 (no cakes sold): P=15(0)−45=−45P = 15(0) - 45 = -45. This means the bakery loses $45 when selling zero cakes, representing fixed costs like rent or utilities that must be paid regardless of sales. The $xx −interceptoccurswhen-intercept occurs when P=0P = 0 (break−evenpoint):(break-even point): 0=15c−450 = 15c - 45 ,so, so 15c=4515c = 45 andand c=3c = 3 $. The bakery needs to sell 3 cakes to cover all costs and achieve zero profit. Choice A correctly identifies both: a $45 loss at zero sales and 3 cakes needed to break even. Choice B incorrectly calls the yy-intercept a "profit" when −45-45 represents a loss. Choice C correctly identifies the 45lossbutdescribesthe45 loss but describes the xx −interceptas"selling3cakesresultsin-intercept as "selling 3 cakes results in 0 profit" rather than recognizing this as the break-even point. Choice D completely misinterprets both intercepts—the yy-intercept isn't the profit per cake (that's the slope of 15), and the xx-intercept isn't a dollar amount. Study tip: For linear business models, the yy-intercept typically represents fixed costs (often negative), while the xx-intercept shows the break-even quantity. Always substitute zero for each variable to find these meaningful points.

Question 17

A water tank initially contains 500 gallons and drains at a rate of 12 gallons per minute. After creating an equation to model the gallons remaining over time, what would be the most appropriate scale for the axes when graphing this relationship for the first 60 minutes?

  1. xx-axis: 0 to 60 by 5s; yy-axis: 0 to 500 by 50s
  2. xx-axis: 0 to 60 by 10s; yy-axis: -220 to 500 by 100s (correct answer)
  3. xx-axis: 0 to 720 by 60s; yy-axis: 0 to 500 by 25s
  4. xx-axis: 0 to 500 by 50s; yy-axis: 0 to 60 by 10s
Explanation: The equation is G=500−12tG = 500 - 12t. At t=60t = 60, G=500−12(60)=−220G = 500 - 12(60) = -220 gallons. To show the complete linear relationship over 60 minutes, the yy-axis must extend below 0 to show when the tank would be empty and beyond. Choice B provides appropriate intervals and range.

Question 18

A student creates the equation y=−13x+4y = -\frac{1}{3}x + 4 to model the relationship between study hours per week xx and GPA points lost yy. What is problematic about this model, and what would be a more appropriate domain?

  1. The negative slope suggests more study decreases GPA loss, but this contradicts logic; domain should be all real numbers
  2. The model suggests infinite study hours are possible; domain should be restricted to 0≤x≤120 ≤ x ≤ 12 for realistic weekly study time
  3. The equation allows negative GPA loss (which means GPA gain), creating impossible values; domain should be 0≤x≤120 ≤ x ≤ 12 (correct answer)
  4. The yy-intercept of 4 suggests students lose 4 GPA points without studying; domain should be x≥0x ≥ 0 since negative study time is impossible
Explanation: The model allows yy to become negative when x>12x > 12 (since y=−13x+4y = -\frac{1}{3}x + 4 gives y<0y < 0 when x>12x > 12). Negative "GPA points lost" would mean "GPA points gained," which contradicts the model's purpose. A domain of 0≤x≤120 ≤ x ≤ 12 keeps y≥0y ≥ 0 and represents realistic study hours.

Question 19

A community garden charges a $15 membership fee plus $3 per square foot for plot rental.

If TT represents total cost and ss represents square feet rented, which statement correctly describes both the equation and an important feature of its graph?

  1. T=15s+3T = 15s + 3; the slope represents the membership fee per square foot
  2. T=3s+15T = 3s + 15; the graph will pass through the origin since no cost when s=0s = 0
  3. T=3s+15T = 3s + 15; the slope of 3 represents the rate of cost increase per square foot (correct answer)
  4. T=15s+3T = 15s + 3; the yy-intercept shows the cost per square foot when no membership is purchased
Explanation: The equation is T=3s+15T = 3s + 15 (variable cost per unit times quantity plus fixed cost). The slope of 3 represents the rate at which total cost increases for each additional square foot rented. Choice A has the coefficients reversed, choice B incorrectly states the graph passes through the origin, and choice D misinterprets the yy-intercept.

Question 20

A smartphone plan costs $40 per month plus $5 for each gigabyte of data used over the 2GB included in the base plan. Which equation correctly models the monthly cost $CC basedontotalgigabytesusedbased on total gigabytes used gg $, and what is the cost for using 6 gigabytes?

  1. C=40+5(g−2)C = 40 + 5(g - 2) for g>2g > 2; the cost for 6 gigabytes is $60 (correct answer)
  2. C=40+5gC = 40 + 5g; the cost for 6 gigabytes is $70
  3. C=40+5gC = 40 + 5g; the cost for 6 gigabytes is $30
  4. C=40+5(g−2)C = 40 + 5(g - 2) for g>2g > 2; the cost for 6 gigabytes is $50
Explanation: When you encounter word problems about costs with different rate structures, you need to carefully identify what's included in the base price versus what incurs additional charges. This smartphone plan includes 2GB in the $40 monthly fee, so you only pay extra for data beyond that threshold. The correct equation is $C=40+5(g−2)C = 40 + 5(g - 2) forfor g>2g > 2 .Here′swhy:thebasecostis. Here's why: the base cost is 40, and the additional charge applies only to gigabytes used over the included 2GB. So if you use gg total gigabytes, you pay extra for (g−2)(g - 2) gigabytes at 5each.For6gigabytes:5 each. For 6 gigabytes: C=40+5(6−2)=40+5(4)=40+20=60C = 40 + 5(6 - 2) = 40 + 5(4) = 40 + 20 = 60 $ dollars. Choice B incorrectly charges 5 for every gigabyte used, ignoring that 2GB are included free. Using $$C = 40 + 5g$$ would mean paying $$40 + 5(6) = 70$$ dollars, which overcharges by 10. Choice C uses the same flawed equation as B but arrives at an impossible cost of $30, which is less than the base monthly fee. Choice D uses the correct equation structure but miscalculates the final answer. The equation C=40+5(g−2)C = 40 + 5(g - 2) is right, but 40+5(4)=6040 + 5(4) = 60, not $50. Study tip: In tiered pricing problems, always identify what's included in the base cost before writing your equation. Look for phrases like "included," "first X units," or "over the limit" to determine where additional charges begin. The expression inside parentheses should represent only the amount subject to extra fees.