Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

Algebra

Algebra Help: Creating And Graphing Two Variable Equations

Review real example questions for Creating And Graphing Two Variable Equations in Algebra.

Question 1

A rectangle has length xxx feet and width (x−4)(x-4)(x−4) feet. Let AAA be the area (in square feet). What equation represents the relationship between AAA and xxx?

  1. A=x−4A = x - 4A=x−4
  2. A=x2−4A = x^2 - 4A=x2−4
  3. A=x(x−4)A = x(x-4)A=x(x−4)
  4. A=2x−8A = 2x - 8A=2x−8
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, we have a rectangle with length x and width (x-4), and we need to find the area A. The area of a rectangle is length times width, so A = x × (x-4). This gives us A = x(x-4), which when expanded would be x² - 4x. This equation lets us calculate the area for any value of x greater than 4! Choice C is correct because it accurately represents the relationship using the area formula: length times width equals x times (x-4). Choice B creates a quadratic equation A = x² - 4, but this misses the multiplication: the area isn't x² minus 4, it's x times the quantity (x-4). When you see a product of two expressions, you need to multiply them together, not just subtract! Remember the difference between operations: when finding area of a rectangle, you multiply length times width. Here, that's x times (x-4), which gives x(x-4) or x² - 4x when expanded. Don't confuse this with simply subtracting 4 from x²!

Question 2

A bike rental shop charges 12torentabikeplus12 to rent a bike plus 12torentabikeplus4 per hour. Let hhh be the number of hours and TTT be the total cost in dollars. Which equation models the relationship between hhh and TTT?

  1. T=12h+4T = 12h + 4T=12h+4
  2. h=4T+12h = 4T + 12h=4T+12
  3. T=4h+12T = 4h + 12T=4h+12
  4. T=4(h+12)T = 4(h + 12)T=4(h+12)
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 hours), choose variables to represent them (like T for total cost and h for hours), then write an equation that captures how one depends on the other. From the context, cost is 12fixedplus12 fixed plus 12fixedplus4 per hour, we identify that T depends on h; the rate is 4 (that becomes our coefficient), and the starting amount is 12 (that's our constant term), so the equation is T = 4h + 12, which lets us calculate total cost for any hours rented! Choice C is correct because it accurately represents the relationship with the fixed fee as the constant and the hourly rate as the coefficient. Choice A has the numbers switched: it uses 12h + 4, but the context tells us 4perhour(not4 per hour (not 4perhour(not12) and 12fixed(not12 fixed (not 12fixed(not4); 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 4perhour=4timeshours),′plus′meansadd(likeplus4 per hour = 4 times hours), 'plus' means add (like plus 4perhour=4timeshours),′plus′meansadd(likeplus12 fee). After you create your equation, test it with simple values: try h = 0 (should give 12)andh=1(shouldgive12) and h = 1 (should give 12)andh=1(shouldgive16); if it matches, you've got it right!

Question 3

A runner moves at a constant speed of 6 miles per hour. Let ddd be the distance (miles) and ttt be the time (hours). Which equation models this relationship?

  1. d=6td = 6td=6t
  2. d=t+6d = t + 6d=t+6
  3. t=6dt = 6dt=6d
  4. d=6+td = 6 + td=6+t
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 runner moves at 6 miles per hour, we identify that distance d depends on time t. The rate is 6 miles per hour (that becomes our coefficient), and there's no starting distance mentioned (so no constant term). So the equation is d = 6t. This equation lets us calculate the distance traveled for any amount of time! Choice A is correct because it accurately represents the relationship with distance equal to 6 times the time, matching the constant speed of 6 miles per hour. Choice C has the variables switched: it says t = 6d, which would mean time equals 6 times the distance, but that would give us a speed of 1/6 miles per hour, not 6 miles per hour. When you see words like 'per,' that usually means multiplication in the direction stated! After you create your equation, test it with simple values: if d = 6t represents distance at 6 mph, try t = 1 (should give 6 miles) and t = 2 (should give 12 miles). If your equation gives the right outputs for these test inputs, you probably have it right!

Question 4

A movie theater charges a 4bookingfeeplus4 booking fee plus 4bookingfeeplus9 for each ticket. Let CCC be the total cost (in dollars) and let ttt be the number of tickets. What equation represents the relationship between ttt and CCC?

  1. C=4tC = 4tC=4t
  2. C=13tC = 13tC=13t
  3. C=9+4tC = 9 + 4tC=9+4t
  4. C=9t+4C = 9t + 4C=9t+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 theater charges 9perticketplusa9 per ticket plus a 9perticketplusa4 booking fee, we identify that total cost C depends on number of tickets t. The rate is 9perticket(thatbecomesourcoefficient),andthebookingfeeis9 per ticket (that becomes our coefficient), and the booking fee is 9perticket(thatbecomesourcoefficient),andthebookingfeeis4 (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 9perticket(9t)plusthe9 per ticket (9t) plus the 9perticket(9t)plusthe4 booking fee (+4). Choice A has the numbers switched: it puts 4perticketanda4 per ticket and a 4perticketanda9 fee, but the context tells us it's 9perticketanda9 per ticket and a 9perticketanda4 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 9perticket=9timesnumberoftickets).′Plus′or′and′meansadd(like9 per ticket = 9 times number of tickets). 'Plus' or 'and' means add (like 9perticket=9timesnumberoftickets).′Plus′or′and′meansadd(like4 fee plus ticket cost). Listen to the language!

Question 5

A streaming service charges a 5sign−upfeeandthen5 sign-up fee and then 5sign−upfeeandthen8 per month. If mmm is the number of months and CCC 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 ($); y-axis: Months
  2. x-axis: Months; y-axis: Total Cost ($)
  3. x-axis: Months ($); y-axis: Total Cost (months)
  4. x-axis: xxx; y-axis: yyy
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 ()′,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()' not just 'C'), and a good scale shows your data without bunching it up or spreading it too thin.

Question 6

A garden has perimeter 40 feet. Let lll be the length (feet) and www be the width (feet). Create an equation relating lll and www that represents this relationship for graphing.

  1. l+w=40l + w = 40l+w=40
  2. 2l+2w=402l + 2w = 402l+2w=40
  3. 2l−2w=402l - 2w = 402l−2w=40
  4. lw=40lw = 40lw=40
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 length and width), choose variables to represent them (like l for length and w for width), then write an equation that captures how one depends on the other. From the context, perimeter is 40 feet which means twice length plus twice width equals 40, so the equation is 2l + 2w = 40; this lets us find possible widths for any length (or vice versa) to maintain the perimeter! Choice B is correct because it accurately represents the relationship with the sum of twice each dimension equaling the perimeter. Choice A has the wrong operation: it uses lw = 40 which is area, but the context says perimeter which means addition of sides; when you see 'perimeter,' think sum of sides! Quick trick: words like 'perimeter' mean add up the sides (2l + 2w), while 'area' means multiply (l * w); listen to the language! When creating equations, test with values: if l=10, then 2*10 + 2w=40 so w=10 (square garden); if it makes sense, you're on track!

Question 7

A school fundraiser sells bracelets for 5eachandchargesaone−time5 each and charges a one-time 5eachandchargesaone−time10 setup fee for the order. Let bbb be the number of bracelets and let CCC be the total cost (in dollars). What equation represents the relationship between CCC and bbb?​

  1. C=5bC = 5bC=5b
  2. C=10−5bC = 10 - 5bC=10−5b
  3. C=10b+5C = 10b + 5C=10b+5
  4. C=5b+10C = 5b + 10C=5b+10
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 5eachplusaone−time5 each plus a one-time 5eachplusaone−time10 setup fee, so the total cost C depends on the number of bracelets b. The rate is 5perbracelet(thatbecomesourcoefficient),andthestartingamountis5 per bracelet (that becomes our coefficient), and the starting amount is 5perbracelet(thatbecomesourcoefficient),andthestartingamountis10 (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 5perbracelet(5b)plusthe5 per bracelet (5b) plus the 5perbracelet(5b)plusthe10 setup fee. Choice A has the numbers switched: it puts 10perbraceletanda10 per bracelet and a 10perbraceletanda5 fee, but the problem clearly states 5isperbraceletand5 is per bracelet and 5isperbraceletand10 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 5perbracelet=5timesnumberofbracelets).′Plus′or′and′meansadd(like5 per bracelet = 5 times number of bracelets). 'Plus' or 'and' means add (like 5perbracelet=5timesnumberofbracelets).′Plus′or′and′meansadd(like10 fee plus bracelet cost). Listen to the language!

Question 8

A streaming service charges 10permonthplusaone−timesetupfeeof10 per month plus a one-time setup fee of 10permonthplusaone−timesetupfeeof5. Let mmm be months and CCC 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
  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(setupfee)andafter6monthsreaches5 (setup fee) and after 6 months reaches 5(setupfee)andafter6monthsreaches5 + 10(6)=10(6) = 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 9

A rectangle has perimeter 40 feet. Let lll be the length (feet) and www be the width (feet). What equation represents the relationship between lll and www?

  1. lw=40lw = 40lw=40
  2. 2l+2w=402l + 2w = 402l+2w=40
  3. l+w=40l + w = 40l+w=40
  4. 2l−2w=402l - 2w = 402l−2w=40
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, a rectangle's perimeter is the distance around it, we identify that perimeter depends on both length l and width w. The perimeter formula is P = 2l + 2w (two lengths plus two widths), and we're told P = 40. So the equation is 2l + 2w = 40. This equation lets us find valid length-width combinations! Choice B is correct because it accurately represents the perimeter relationship 2l + 2w = 40. Choice A creates an equation lw = 40, but that would be the area formula (length times width), not perimeter. When you see 'perimeter,' that means the distance around, which requires adding all sides! When creating equations from word problems, ask yourself: What formula applies here? Perimeter of a rectangle is 2l + 2w (add all four sides), while area is l × w (multiply dimensions). Don't mix them up!

Question 10

A video game store sells used games for 12each.Let12 each. Let 12each.Letnbethenumberofgamesandbe the number of games andbethenumberofgamesandC$ be the total cost (in dollars). When graphing this relationship, what should the axes be labeled?

  1. x-axis: Total Cost ($), y-axis: Number of Games
  2. x-axis: Number of Games, y-axis: Total Cost ($)
  3. x-axis: Games ($), y-axis: Cost (number)
  4. x-axis: xxx, y-axis: yyy
Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. The coordinate plane helps us visualize relationships: the horizontal x-axis typically shows the independent variable (the one you choose or that changes first, like time or quantity), while the vertical y-axis shows the dependent variable (the one that responds, like cost or height). For this relationship, the x-axis should represent Number of Games with that label, and the y-axis should represent Total Cost ($) with that label. This makes sense because you choose how many games to buy (independent), and the cost depends on that choice (dependent). Choice B is correct because it sets up the axes appropriately with the independent variable (number of games) on the x-axis and the dependent variable (total cost) on the y-axis, including proper units. Choice A has the variables switched: it puts Total Cost on the x-axis and Number of Games 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 items you buy), 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!