- A scooter travels at a constant speed of 12 miles per hour. Let d be the distance (miles) and h be the time (hours). Which equation models this relationship?
Answer: d=12h
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 scooter travels at 12 miles per hour, we identify that distance d depends on time h. The rate is 12 miles per hour (that becomes our coefficient), and there's no starting distance mentioned (so no constant term). So the equation is d = 12h. 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 equals rate times time (d = 12h). Choice B has the variables switched: it suggests time equals 12 times distance, but that would mean hours = 12 × miles, which doesn't make sense. When you see 'miles per hour,' that's a rate that multiplies time to give distance! After you create your equation, test it with simple values: if d = 12h represents distance at 12 mph, try h = 1 (should give 12 miles) and h = 2 (should give 24 miles). If your equation gives the right outputs for these test inputs, you probably have it right! - A music streaming service charges $3 per month plus a one-time $10 setup fee. Let m be the number of months and let C be the total cost (in dollars). Which equation models this context?
Answer: C=3m+10
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 service charges $3 per month plus a one-time $10 setup fee, we identify that total cost C depends on number of months m. The rate is $3 per month (that becomes our coefficient), and the setup fee is $10 (that's our constant term). So the equation is C = 3m + 10. This equation lets us calculate the total cost for any number of months! Choice C is correct because it accurately represents the relationship with $3 per month (3m) plus the $10 setup fee (+10). Choice A has the numbers switched: it puts $10 per month and a $3 fee, but the context tells us it's $3 per month and a $10 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 $3 per month = 3 times number of months). 'Plus' or 'one-time fee' means add (like $10 setup fee plus monthly cost). Listen to the language! - A student creates the equation y=−31x+4 to model the relationship between study hours per week x and GPA points lost y. What is problematic about this model, and what would be a more appropriate domain?
Answer: The equation allows negative GPA loss (which means GPA gain), creating impossible values; domain should be 0≤x≤12
The model allows y to become negative when x>12 (since y=−31x+4 gives y<0 when x>12). Negative "GPA points lost" would mean "GPA points gained," which contradicts the model's purpose. A domain of 0≤x≤12 keeps y≥0 and represents realistic study hours. - A taxi charges a $3 start fee plus $2.50 per mile. Let m be miles traveled and C be the total cost (in dollars). Which equation models this context?
Answer: C=2.5m+3
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 $3 to start (a fixed fee) plus $2.50 per mile, so the total cost C depends on the number of miles m. The rate is $2.50 per mile (that becomes our coefficient), and the starting amount is $3 (that's our constant term). So the equation is C = 2.5m + 3. This equation lets us calculate the total cost for any trip distance! Choice C is correct because it accurately represents the relationship with $2.50 per mile (2.5m) plus the $3 start fee. Choice B has the numbers switched: it puts $3 per mile and a $2.50 fee, but the problem clearly states $3 is the start fee and $2.50 is per mile. When translating words to equations, make sure each part of the equation corresponds to something in the description! After you create your equation, test it with simple values: if C = 2.5m + 3 represents cost for m miles, try m = 0 (should give the $3 start fee) and m = 2 (should give $8). If your equation gives the right outputs for these test inputs, you probably have it right! - A streaming service charges a $5 sign-up fee and then $8 per month. If m is the number of months and C is the total cost (in dollars), what should the axes be labeled when graphing this relationship on a coordinate plane?
Answer: x-axis: Months; y-axis: Total Cost ($)
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()' not just 'C'), and a good scale shows your data without bunching it up or spreading it too thin. - A relationship is described by: "y is 4 less than three times x." What equation represents this relationship?
Answer: y=3x−4
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! - A gym membership charges a $25 sign-up fee plus $10 per month. Let p be the number of months and C be total cost in dollars. What should the axes be labeled when graphing this relationship?
Answer: x-axis: Months (months), y-axis: Cost ($)
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 months), while the vertical y-axis shows the dependent variable (the one that responds, like cost). For this relationship, the x-axis should represent months with label 'Months (months)', and the y-axis should represent total cost with label 'Cost ($)'; this makes sense because months is the independent variable (you choose how many), and cost depends on that including the sign-up fee. Choice B is correct because it sets up the axes appropriately with months on x and cost on y, including correct units. Choice A has the variables switched: it puts cost on the x-axis and months on the y-axis, but remember—the independent variable goes on the x-axis, and the dependent on the y-axis. For graphing, think about your audience: good axis labels include the variable name AND units (like 'Months (months)' not just 'p'), and this clarity helps everyone see the relationship quickly. Remember the difference between independent and dependent variables: independent is what you control (like months), dependent is the outcome (like cost)—stick to x for independent, y for dependent!
- A rectangle has perimeter 40 feet. Let l be the length (feet) and w be the width (feet). What equation represents the relationship between l and w?
Answer: 2l+2w=40
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! - A rectangle has length x feet and width (x−4) feet. Let A be the area (in square feet). What equation represents the relationship between A and x?
Answer: A=x(x−4)
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! - A garden has perimeter 40 feet. Let l be the length (feet) and w be the width (feet). Create an equation relating l and w that represents this relationship for graphing.
Answer: 2l+2w=40
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! - A gym charges a one-time sign-up fee of $25 and then $15 per month. Let m be the number of months and let C be the total cost (in dollars). Which equation models the total cost?
Answer: C=15m+25
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! - 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?
Answer: x-axis: 0 to 60 by 10s; y-axis: -220 to 500 by 100s
The equation is G=500−12t. At t=60, G=500−12(60)=−220 gallons. To show the complete linear relationship over 60 minutes, the y-axis must extend below 0 to show when the tank would be empty and beyond. Choice B provides appropriate intervals and range. - A student earns $8 per hour for babysitting and already has $15 saved. Let h be hours worked and M be total money (in dollars). If graphing this relationship for up to 10 hours, how should the coordinate plane be set up to show the context?
Answer: x-axis: Hours (hours); y-axis: Total Money ($)
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 hours worked with label 'Hours (hours)', and the y-axis should represent total money with label 'Total Money ($)'. This makes sense because you choose how many hours to work (independent), and your total money depends on that choice (dependent). Choice B is correct because it sets up the axes appropriately with hours on the x-axis (the independent variable you control) and total money on the y-axis (the dependent variable that responds). Choice A has the variables switched: it puts Total Money on the x-axis and Hours 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 hours you work), and the dependent variable is the one that responds to your choice (like what your total money is). Independent goes on the x-axis, dependent on the y-axis—this is the standard convention!
- A car rental costs $30 plus $0.20 per mile driven. Let m be miles driven and C be total cost in dollars. How should the coordinate plane be set up to show this context?
Answer: x-axis: Miles (miles), y-axis: Cost ($)
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 miles driven), while the vertical y-axis shows the dependent variable (the one that responds, like cost). For this relationship, the x-axis should represent miles driven with label 'Miles (miles)', and the y-axis should represent total cost with label 'Cost ($)'; this makes sense because miles is the independent variable (you choose how far to drive), and cost depends on that choice. Choice B is correct because it sets up the axes appropriately with miles on x and cost on y, including correct units. Choice A has the variables switched: it puts cost on the x-axis and miles on the y-axis, but remember—the independent variable goes on the x-axis, and the dependent on the y-axis. Remember the difference between independent and dependent variables: independent is what you control (like miles), dependent is what results (like cost); x for independent, y for dependent! For graphing, think about your audience: good axis labels include the variable name AND units (like 'Miles (miles)' not just 'm'), so anyone can understand right away.
- A movie theater charges a $6 admission fee plus $2 for each snack you buy. Let C be the total cost (in dollars) and let s be the number of snacks. Write an equation that represents the relationship between C and s.
Answer: C=2s+6
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 movie theater charges $6 admission (a fixed fee) plus $2 for each snack, so the total cost C depends on the number of snacks s. The rate is $2 per snack (that becomes our coefficient), and the starting amount is $6 (that's our constant term). So the equation is C = 2s + 6. This equation lets us calculate the total cost for any number of snacks! Choice B is correct because it accurately represents the relationship with $2 per snack (2s) plus the $6 admission fee. Choice A has the numbers switched: it puts $6 per snack and a $2 fee, but the problem clearly states $6 is the admission fee and $2 is per snack. 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 snack = 2 times number of snacks). 'Plus' means add (like $6 fee plus snack cost). Listen to the language! - A school club has $120 to spend on supplies. They plan to buy x notebooks that cost $3 each and y packs of markers that cost $6 each. Which equation represents all combinations of x and y that use exactly 120?
Answer: 3x+6y=120
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, notebooks cost $3 each and markers cost $6 per pack, with a total budget of $120, we identify that the total cost depends on both x notebooks and y marker packs. The cost of notebooks is 3x (rate times quantity), the cost of markers is 6y (rate times quantity), and together they must equal $120. So the equation is 3x + 6y = 120. This equation lets us find all combinations of notebooks and markers that use exactly the budget! Choice A is correct because it accurately represents the relationship with 3x as the total cost of notebooks, 6y as the total cost of markers, and their sum equaling the $120 budget. Choice D has the coefficients switched: it makes notebooks cost $6 and markers $3, but the problem clearly states notebooks are $3 and markers are $6. When translating words to equations, make sure each coefficient matches its corresponding item! When creating equations from word problems, ask yourself three questions: (1) What are the two quantities that are related? (2) What's the cost or value of each? (3) What's the total constraint? Here we have two items with different costs that must add up to a fixed total—this creates a linear equation in two variables! - A taxi charges a $4 starting fee plus $2 per mile. Let m be miles traveled and C be total cost (in dollars). Which equation represents this relationship?
Answer: C=2m+4
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! - A student earns $8 per hour babysitting and starts with $0 earned. Let h be hours worked and E be total earnings in dollars. What is an appropriate scale for graphing this relationship if the student might work up to 10 hours?
Answer: x-axis: 0 to 10 by 1; y-axis: 0 to 80 by 10
This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. For graphing an equation, choose a scale that fits your data: if your values range from 0 to 50, marking every 5 or 10 units makes sense, but marking every 1 would be cramped; a good scale makes the graph readable and shows the pattern clearly. Looking at the context, hours worked ranges from 0 to 10, so a good scale for the x-axis would be marking every 1 unit; similarly, earnings range from 0 to 80 (at $8/hour), so marking every 10 on y-axis works well; this scale shows the data clearly without cramming or spreading too thin! Choice A is correct because it chooses a reasonable scale with x from 0 to 10 by 1 (fits up to 10 hours) and y from 0 to 80 by 10 (fits up to $80). Choice B uses a scale that's not practical: with hours up to 10, x from 0 to 100 by 10 is too spread out, and y to 10 by 1 is too small for $80 max; a better scale matches the data range. For graphing, think about your audience: a good scale shows your data without bunching it up or spreading it too thin; if values go to 10, try every 1 or 2—not every 10 (too sparse) or every 0.1 (too crowded). Remember, the goal is to make the linear pattern (like earnings increasing steadily) easy to see and interpret!
- A bakery's daily profit P depends on the number of specialty cakes c sold according to P=15c−45. When graphing this relationship, what do the intercepts represent in this business context?
Answer: y-intercept: $45 loss when no cakes sold; x-intercept: 3 cakes needed to break even
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−45, let's find both intercepts. The y-intercept occurs when c=0 (no cakes sold): P=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 x-intercept occurs when P=0 (break-even point): 0=15c−45, so 15c=45 and c=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 y-intercept a "profit" when −45 represents a loss. Choice C correctly identifies the $45 loss but describes the x-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 y-intercept isn't the profit per cake (that's the slope of 15), and the x-intercept isn't a dollar amount.
Study tip: For linear business models, the y-intercept typically represents fixed costs (often negative), while the x-intercept shows the break-even quantity. Always substitute zero for each variable to find these meaningful points. - A car travels at a constant speed of 55 miles per hour. Let t be time (hours) and let d be distance (miles). If you want to graph this relationship for 0≤t≤4, what is an appropriate scale for the coordinate plane?
Answer: x-axis (time): 0 to 4 by 1; y-axis (distance): 0 to 240 by 60
This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. For graphing an equation, choose a scale that fits your data: if your values range from 0 to 50, marking every 5 or 10 units makes sense, but marking every 1 would be cramped. A good scale makes the graph readable and shows the pattern clearly. Looking at the context, time ranges from 0 to 4 hours, so a good scale for that axis would be marking every 1 hour. For distance, at 55 mph for 4 hours gives us 220 miles maximum, so marking every 60 miles works well (giving us 0, 60, 120, 180, 240). 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 time from 0 to 4 by 1 (perfect for our 4-hour range) and distance from 0 to 240 by 60 (which nicely accommodates our maximum of 220 miles). Choice B uses a scale that's not practical: it has time going to 40 hours when we only need 4, and distance only to 4 miles when we need up to 220. A better scale is what Choice A provides, which shows the pattern 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 220, try marking every 50 or 60—not every 1 (too crowded) or every 1000 (too sparse).
