Road to 800

Linear word problems

Chapter 4 · 8 lessons · 484 questions

Writing an equation from a story

Key points

  • The constant is the fixed or starting amount. It is counted once.
  • The unknown number is a rate times the variable: so much per item, per hour or per month.
  • The ending number is the total.
  • Constant plus rate times variable equals total. If the amount goes down, subtract instead of add.
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See it solved

Example

A plumber charges a call-out fee of $45\$45 plus $60\$60 for each hour of work. The plumber charges $285\$285 for one job. Which equation gives hh, the number of hours the job took?

  1. A45h+60=28545h + 60 = 285
  2. B60h+45=28560h + 45 = 285
  3. C60h−45=28560h - 45 = 285
  4. D105h=285105h = 285

Worked by hand

The plan

  1. Determine the constant
  2. Determine the unknown number
  3. Determine the ending number

Desmos does not write the equation for you. It only solves the finished equation, as a check.

  1. Step 1: Determine the constant

    • The call-out fee, $45\$45, is charged once. It is the constant.
    • That rules out choice A, which multiplies 4545 by hh, and choice D, which has no constant.

  2. Step 2: Determine the unknown number

    • The work costs $60\$60 for each hour, for hh hours: 60h60h dollars.

  3. Step 3: Determine the ending number

    • The total is $285\$285. The fee is added to the work, not taken away.
    • 60h+45=28560h + 45 = 285. That rules out choice C.
    • Check by solving. Subtract 4545 from both sides: 60h60h =285−45= 285 - 45 =240= 240.
    • Divide both sides by 6060: h=24060=4h = \dfrac{240}{60} = 4 hours.
    • 44 hours of work cost 60⋅4=24060 \cdot 4 = 240 dollars, and 240+45=285240 + 45 = 285. It matches.
The correct answer choice is B: 60h+45=28560h + 45 = 285.

The same example in Desmos

  1. new graph

    A vertical line, with xx for the hours.

  2. click where the line meets the x-axis

    (4,0)(4, 0): the job took 44 hours, which fits the story.

Type
Desmos expression list: 60x+45=285
Look
Desmos graph of 60x+45=285, with (4, 0) labelled

Made with Desmos

Open Desmos

Your turn

A tank holds 500500 liters of water. Water drains out at 3535 liters per minute. After tt minutes, 150150 liters remain. What is the value of tt?

Show the answerHide the answer

Answer

The correct answer is 1010.

  1. Step 1: Determine the constant

    • The tank starts with 500500 liters. That is the constant.

  2. Step 2: Determine the unknown number

    • 3535 liters drain each minute, for tt minutes: 35t35t liters.
    • The water goes down, so this part is subtracted.

  3. Step 3: Determine the ending number

    • 150150 liters remain: 500−35t=150500 - 35t = 150.
    • Subtract 500500 from both sides: −35t-35t =150−500= 150 - 500 =−350= -350.
    • Divide both sides by −35-35: t=−350−35=10t = \dfrac{-350}{-35} = 10.

Traps

  • Putting the rate on the wrong number. 45h+60=28545h + 60 = 285 charges the call-out fee every hour. That is choice A, a wrong answer.
  • Adding the two numbers before multiplying. 105h105h also charges the fee every hour. That is choice D, a wrong answer.
Practice this lesson64 questions

Two amounts that make a total

Key points

  • Two parts add up to a total: ax+by=cax + by = c.
  • Each part is a rate times its own variable: price times how many, or speed times time.
  • The total stands by itself on the other side.
  • If one amount is known, plug it in and solve for the other.
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See it solved

Example

A bakery sells muffins for $3\$3 each and cookies for $2\$2 each. One day it sells xx muffins and yy cookies for a total of $126\$126. It sells 2424 muffins that day. How many cookies does it sell?

Worked by hand

The plan

  1. Determine the unknown numbers
  2. Determine the ending number
  3. Determine the value of yy

Desmos does not help you write the equation. Once it is written, typing it is only a check.

  1. Step 1: Determine the unknown numbers

    • Muffins: $3\$3 each, and xx of them. That part is 3x3x dollars.
    • Cookies: $2\$2 each, and yy of them. That part is 2y2y dollars.

  2. Step 2: Determine the ending number

    • The two parts add up to the total, $126\$126.
    • 3x+2y=1263x + 2y = 126.

  3. Step 3: Determine the value of yy

    • Plug in x=24x = 24: 3⋅24+2y3 \cdot 24 + 2y =126= 126.
    • 3⋅24=723 \cdot 24 = 72, so 72+2y=12672 + 2y = 126.
    • Subtract 7272 from both sides: 2y2y =126−72= 126 - 72 =54= 54.
    • Divide both sides by 22: y=542=27y = \dfrac{54}{2} = 27.
    • Check: 3⋅24+2⋅273 \cdot 24 + 2 \cdot 27 =72+54= 72 + 54 =126= 126. It matches.
The correct answer is 2727.

The same example in Desmos

  1. new graph

    A vertical line. Here xx stands for the number of cookies.

  2. click where the line meets the x-axis

    (27,0)(27, 0): 2727 cookies.

Type
Desmos expression list: 3(24)+2x=126
Look
Desmos graph of 3(24)+2x=126, with (27, 0) labelled

Made with Desmos

Open Desmos

Your turn

A printer takes 55 seconds to print a color page and 22 seconds to print a black-and-white page. One job has cc color pages and bb black-and-white pages, and it takes 160160 seconds. The job has 4040 black-and-white pages. How many color pages does it have?

Show the answerHide the answer

Answer

The correct answer is 1616.

  1. Step 1: Determine the unknown numbers

    • Color pages: 55 seconds each, and cc of them. That part is 5c5c seconds.
    • Black-and-white pages: 22 seconds each, and bb of them. That part is 2b2b seconds.

  2. Step 2: Determine the ending number

    • The two parts add up to the total, 160160 seconds.
    • 5c+2b=1605c + 2b = 160.

  3. Step 3: Determine the value of cc

    • Plug in b=40b = 40: 5c+2⋅405c + 2 \cdot 40 =160= 160.
    • 2⋅40=802 \cdot 40 = 80, so 5c+80=1605c + 80 = 160.
    • Subtract 8080 from both sides: 5c5c =160−80= 160 - 80 =80= 80.
    • Divide both sides by 55: c=805=16c = \dfrac{80}{5} = 16.

Traps

  • Swapping the two rates. 2x+3y=1262x + 3y = 126 puts the cookie price on the muffins. Then 2⋅24=482 \cdot 24 = 48, 126−48=78126 - 48 = 78 and y=783=26y = \dfrac{78}{3} = 26, a wrong answer.
  • Adding the counts instead of the costs. x+y=126x + y = 126 counts items, not dollars. It gives yy =126−24= 126 - 24 =102= 102, a wrong answer.
Practice this lesson48 questions

Building a linear model

Key points

  • A linear model is a starting amount plus the same change for each step: y=mx+by = mx + b.
  • The constant, bb, is the fixed or starting amount: the value when x=0x = 0.
  • The slope, mm, is the change for each step of 11 in xx. Its units have ‘per’ in them, like dollars per month.
  • Two situations in a story give you two points: (input, output).
  • A negative slope is a steady drop: −4-4 means ‘down 44 each time’.
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See it solved

Example

A gym charges a joining fee plus the same fee each month. After 22 months, a member has paid $90\$90 in all. After 66 months, the total is $190\$190. Write a linear function C(n)C(n) for the total cost, in dollars, after nn months.

Worked by hand

The plan

  1. Determine the slope
  2. Determine the constant

With messy numbers, Desmos is quicker: put the two situations in a table and fit a line.

  1. Step 1: Determine the slope

    • Each situation is a point (months, dollars): (2,90)(2, 90) and (6,190)(6, 190).
    • The months go up by 6−2=46 - 2 = 4.
    • The total goes up by 190−90=100190 - 90 = 100.
    • m=1004=25m = \dfrac{100}{4} = 25. That is $25\$25 per month.

  2. Step 2: Determine the constant

    • Plug (2,90)(2, 90) and m=25m = 25 into C=mn+bC = mn + b.
    • 90=25⋅2+b90 = 25 \cdot 2 + b.
    • 25⋅2=5025 \cdot 2 = 50, so 90=50+b90 = 50 + b.
    • b=90−50=40b = 90 - 50 = 40. That is the joining fee, $40\$40.
    • Check: 66 months gives 25⋅6+4025 \cdot 6 + 40 =150+40= 150 + 40 =190= 190. It matches.
    • The model is C(n)=25n+40C(n) = 25n + 40.
The correct answer is C(n)=25n+40C(n) = 25n + 40.

The same example in Desmos

  1. new graph, then Add item + and Table

    Table
    x1y1
    290
    6190

    The two situations as points (months, dollars).

  2. Keys: y1~mx1+b

    m=25m = 25 and b=40b = 40.

  3. The model's line, through both points.

  4. click the line, then its gray point on the y-axis

    (0,40)(0, 40): at 00 months the cost is $40\$40, the joining fee.

Type
Desmos expression list: a table of x1: 2, 6 and y1: 90, 190; y1~ mx1+b; y=mx+b
Look
Desmos graph of a table of x1: 2, 6 and y1: 90, 190; y1~ mx1+b; y=mx+b, with (0, 40) labelled

Made with Desmos

Open Desmos

Your turn

A tank drains at a constant rate. After 33 minutes, it holds 7070 gallons of water. After 77 minutes, it holds 3030 gallons. Write a linear function V(t)V(t) for the gallons in the tank after tt minutes.

Show the answerHide the answer

Answer

The correct answer is V(t)=100−10tV(t) = 100 - 10t.

  1. Step 1: Determine the slope

    • Points (minutes, gallons): (3,70)(3, 70) and (7,30)(7, 30).
    • The minutes go up by 7−3=47 - 3 = 4.
    • The water changes by 30−70=−4030 - 70 = -40. It drops 4040 gallons.
    • m=−404=−10m = \dfrac{-40}{4} = -10. The tank loses 1010 gallons each minute.

  2. Step 2: Determine the constant

    • Plug (3,70)(3, 70) and m=−10m = -10 into V=mt+bV = mt + b.
    • 70=−10⋅3+b70 = -10 \cdot 3 + b.
    • −10⋅3=−30-10 \cdot 3 = -30, so 70=−30+b70 = -30 + b.
    • bb =70+30= 70 + 30 =100= 100. The tank held 100100 gallons at the start.
    • V(t)V(t) =−10t+100= -10t + 100. That is the same as 100−10t100 - 10t.

Traps

  • Dividing one total by its months. 90÷2=4590 \div 2 = 45, but the $90\$90 includes the joining fee, so 4545 is not the monthly fee. That slip gives C(n)=45nC(n) = 45n, a wrong answer. Use the change between two situations.
  • Losing the minus sign. When the amount goes down, the change is negative: 30−70=−4030 - 70 = -40, not 4040. Using 4040 gives a slope of 40÷4=1040 \div 4 = 10. Then 70=10⋅3+b70 = 10 \cdot 3 + b. 10⋅3=3010 \cdot 3 = 30, so b=70−30=40b = 70 - 30 = 40. That gives V(t)=10t+40V(t) = 10t + 40, a wrong answer.
Practice this lesson80 questions

What the numbers in a model mean

Key points

  • The slope, mm, is the change in yy for each 11 more of xx. Its units have ‘per’ in them, like dollars per hour.
  • The constant, bb, is the value of yy when x=0x = 0: a fixed fee, or a starting amount.
  • A negative slope means the amount goes down: −3-3 means ‘33 less each time’.
  • When a model is fitted to data, the slope is an estimate: the average change, not an exact one.
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See it solved

Example

A bike shop charges C(h)=6h+15C(h) = 6h + 15 dollars to rent a bike for hh hours. What does 66 mean? What does 1515 mean?

Worked by hand

The plan

  1. Determine the components of the equation

Desmos does not help you decide what a number means. Use it only as a check.

  1. Step 1: Determine the components of the equation

    • C(h)C(h) is the total cost, in dollars. hh is the number of hours.
    • 6h6h is 66 times the hours: the part that grows with each hour.
    • 1515 is added once, whatever the hours.
    • Plug in h=0h = 0: C(0)C(0) =6⋅0+15= 6 \cdot 0 + 15 =0+15= 0 + 15 =15= 15.
    • So 1515 is the cost at 00 hours: a fee, paid once.
    • Plug in h=1h = 1: C(1)C(1) =6⋅1+15= 6 \cdot 1 + 15 =6+15= 6 + 15 =21= 21.
    • 21−15=621 - 15 = 6. So each hour adds $6\$6.
    • 66 is in dollars per hour: the cost for each hour.
66 is the cost for each hour: $6\$6 per hour. 1515 is a fee paid once: $15\$15.

The same example in Desmos

  1. new graph

    The model, with xx for the hours.

  2. 1515, at the right of the row: the cost at 00 hours.

  3. 66: what one more hour adds.

Type
Desmos expression list: f(x)=6x+15; f(0); f(1)-f(0)
Look
Desmos graph of f(x)=6x+15; f(0); f(1)-f(0), with (0, 15) labelled

Made with Desmos

Open Desmos

Your turn

A candle is lit. Its height, in centimeters, tt hours later is H(t)=20−2tH(t) = 20 - 2t. What does 2020 mean? What does the 22 in 2t2t mean?

Show the answerHide the answer

Answer

2020 is the height when the candle is lit: 2020 centimeters. 22 is how much shorter it gets each hour: 22 centimeters per hour.

  1. Step 1: Determine the components of the equation

    • H(t)H(t) is the height, in centimeters. tt is the hours since the candle was lit.
    • Plug in t=0t = 0: H(0)H(0) =20−2⋅0= 20 - 2 \cdot 0 =20−0= 20 - 0 =20= 20.
    • So 2020 is the height at the start: 2020 centimeters.
    • Plug in t=1t = 1: H(1)H(1) =20−2⋅1= 20 - 2 \cdot 1 =20−2= 20 - 2 =18= 18.
    • 18−20=−218 - 20 = -2. So each hour, the candle gets 22 centimeters shorter.
    • The slope is −2-2, in centimeters per hour.

Traps

  • Swapping the two numbers. 1515 has no ‘per hour’ in it, so it is not the cost per hour. The number with ‘per’ in its units is the slope; the value at 00 hours is the constant. ‘$15\$15 per hour’ is a wrong answer.
  • The fee is not the cost of the first hour. The first hour costs 6⋅1+156 \cdot 1 + 15 =6+15= 6 + 15 =21= 21 dollars. 1515 is the cost at 00 hours: the fee alone. ‘The cost of the first hour’ is a wrong answer for 1515.
Practice this lesson64 questions

Systems from word problems

Key points

  • Two unknowns need two equations.
  • One equation counts things: how many in all.
  • The other equation adds values: the cost, the points or the weight in all.
  • Say in words what each variable stands for before you write anything.
  • Answer what the question asks. It may want only one of the two unknowns.
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See it solved

Example

A theater sold 200200 tickets for a show. Adult tickets cost $12\$12 each and student tickets cost $8\$8 each. The theater collected $2,080\$2{,}080 in all. How many adult tickets were sold?

Worked by hand

The plan

  1. Identify the two variables
  2. Determine the system of equations
  3. Solve the system of equations

Write the two equations by hand. Then Desmos is quicker for Step 3: type both and click the crossing.

  1. Step 1: Identify the two variables

    • Let aa be the number of adult tickets.
    • Let ss be the number of student tickets.

  2. Step 2: Determine the system of equations

    • Count the tickets: a+s=200a + s = 200.
    • Add the money: 12a+8s=2,08012a + 8s = 2{,}080.

  3. Step 3: Solve the system of equations

    • Multiply the first equation by 88: 8a+8s=1,6008a + 8s = 1{,}600.
    • Subtract it from the second equation.
    • 12a−8a=4a12a - 8a = 4a and 8s−8s=08s - 8s = 0.
    • 2,080−1,6002{,}080 - 1{,}600 =480= 480.
    • 4a=4804a = 480, so a=4804=120a = \dfrac{480}{4} = 120.
    • Check: ss =200−120= 200 - 120 =80= 80. 12⋅120=1,44012 \cdot 120 = 1{,}440 and 8⋅80=6408 \cdot 80 = 640.
    • 1,440+6401{,}440 + 640 =2,080= 2{,}080. It matches.
The correct answer is 120120.

The same example in Desmos

  1. new graph

    The count, with xx for adult tickets and yy for student tickets.

  2. The money.

  3. zoom out until the lines cross, then click the crossing

    (120,80)(120, 80): 120120 adult tickets and 8080 student tickets.

Type
Desmos expression list: x+y=200; 12x+8y=2080
Look
Desmos graph of x+y=200; 12x+8y=2080, with (120, 80) labelled

Made with Desmos

Open Desmos

Your turn

A coffee shop sold 5050 drinks one morning. A small drink costs $3\$3 and a large drink costs $5\$5. The shop collected $190\$190 in all. How many large drinks did it sell?

Show the answerHide the answer

Answer

The correct answer is 2020.

  1. Step 1: Identify the two variables

    • Let ss be the number of small drinks.
    • Let ll be the number of large drinks.

  2. Step 2: Determine the system of equations

    • Count the drinks: s+l=50s + l = 50.
    • Add the money: 3s+5l=1903s + 5l = 190.

  3. Step 3: Solve the system of equations

    • Multiply the first equation by 33: 3s+3l=1503s + 3l = 150.
    • Subtract it from the second equation.
    • 3s−3s=03s - 3s = 0 and 5l−3l=2l5l - 3l = 2l.
    • 190−150=40190 - 150 = 40.
    • 2l=402l = 40, so l=402=20l = \dfrac{40}{2} = 20.

Traps

  • Swapping the prices. 8a+12s=2,0808a + 12s = 2{,}080 puts the student price on the adult tickets. With a+s=200a + s = 200, it gives a=80a = 80, a wrong answer.
  • Answering with the other variable. s=80s = 80 is the number of student tickets. The question asks for adult tickets: 120120.
Practice this lesson104 questions

Inequalities from word problems

Key points

  • ‘At least’ means ≥\ge. ‘At most’ and ‘no more than’ mean ≤\le.
  • ‘More than’ means >>. ‘Fewer than’ and ‘less than’ mean <<.
  • Write it like an equation from a story: a constant, plus a rate times the variable, compared with the limit.
  • When the answer counts whole things, round down for a maximum and round up for a minimum.
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See it solved

Example

Maya has $150\$150 to spend at a fair. Entry costs $18\$18, and each ride ticket costs $7\$7. What is the greatest number of ride tickets she can buy?

Worked by hand

The plan

  1. Determine the variables
  2. Determine the conditional relationship
  3. Determine the condition

Desmos does not help you choose the symbol or round the answer. Use it only to check the boundary value.

  1. Step 1: Determine the variables

    • Let tt be the number of ride tickets.

  2. Step 2: Determine the conditional relationship

    • Entry is $18\$18, paid once. The tickets cost 7t7t dollars.
    • She can spend at most $150\$150: 18+7t≤15018 + 7t \le 150.

  3. Step 3: Determine the condition

    • Subtract 1818 from both sides: 7t≤150−187t \le 150 - 18, so 7t≤1327t \le 132.
    • Divide both sides by 77: t≤1327t \le \dfrac{132}{7}.
    • 7⋅18=1267 \cdot 18 = 126 and 7⋅19=1337 \cdot 19 = 133. So 1327\dfrac{132}{7} is between 1818 and 1919.
    • Tickets are whole, and this is a maximum, so round down: 1818.
    • Check: 1818 tickets cost 18+126=14418 + 126 = 144 dollars, which she has. 1919 tickets cost 18+133=15118 + 133 = 151 dollars, which is too much.
The correct answer is 1818.

The same example in Desmos

  1. new graph

    A vertical line, with xx for the tickets.

  2. zoom out, then click where the line meets the x-axis

    About (18.857,0)(18.857, 0). She cannot buy part of a ticket, and 1919 costs too much, so the answer is 1818.

Type
Desmos expression list: 18+7x=150
Look
Desmos graph of 18+7x=150, with (18.857142857142858, 0) labelled

Made with Desmos

Open Desmos

Your turn

A club needs to raise at least $400\$400. It already has $85\$85, and it earns $12\$12 for each shirt it sells. What is the least number of shirts the club must sell?

Show the answerHide the answer

Answer

The correct answer is 2727.

  1. Step 1: Determine the variables

    • Let ss be the number of shirts.

  2. Step 2: Determine the conditional relationship

    • The club has $85\$85 already. The shirts bring in 12s12s dollars.
    • It needs at least $400\$400: 85+12s≥40085 + 12s \ge 400.

  3. Step 3: Determine the condition

    • Subtract 8585 from both sides: 12s≥400−8512s \ge 400 - 85, so 12s≥31512s \ge 315.
    • Divide both sides by 1212: s≥31512s \ge \dfrac{315}{12}.
    • 12⋅26=31212 \cdot 26 = 312 and 12⋅27=32412 \cdot 27 = 324. So 31512\dfrac{315}{12} is between 2626 and 2727.
    • Shirts are whole, and this is a minimum, so round up: 2727.
    • Check: 2626 shirts give 85+312=39785 + 312 = 397 dollars, which is not enough. 2727 shirts give 85+324=40985 + 324 = 409 dollars, which is enough.

Traps

  • Rounding to the nearest whole number. 1327\dfrac{132}{7} is about 18.8618.86, which rounds to 1919. But 1919 tickets cost 18+133=15118 + 133 = 151 dollars, more than she has. 1919 is a wrong answer.
  • Leaving out the fixed cost. 7t≤1507t \le 150 gives t≤21.4t \le 21.4, so 2121 tickets. But the entry fee uses up $18\$18 first. 2121 is a wrong answer.
Practice this lesson80 questions

Constraints in words, and the best corner

Key points

  • Each limit in the story is its own inequality.
  • A limit on how many: x+y≤x + y \le the number allowed.
  • A limit on money or time: each price times its own variable, added, compared with the budget or the goal.
  • The most or the least of something is found at a corner, where two boundary lines cross.
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See it solved

Example

A food truck sells tacos for $4\$4 each and burritos for $7\$7 each. It can make at most 120120 items in a day, and it wants at least $600\$600 in sales. On a day when it sells exactly 120120 items, what is the least number of burritos it must sell to reach its goal?

Worked by hand

The plan

  1. Identify the two variables
  2. Determine the inequality for each constraint
  3. Solve the two boundaries at the corner

Write the inequalities by hand. Then Desmos is quicker for the corner: graph the two boundary lines and click where they cross.

  1. Step 1: Identify the two variables

    • Let tt be the number of tacos.
    • Let bb be the number of burritos.

  2. Step 2: Determine the inequality for each constraint

    • Items: ‘at most 120120’ means ≤\le. So t+b≤120t + b \le 120.
    • Sales: tacos bring in 4t4t dollars and burritos bring in 7b7b dollars.
    • ‘At least $600\$600’ means ≥\ge. So 4t+7b≥6004t + 7b \ge 600.

  3. Step 3: Solve the two boundaries at the corner

    • The truck sells exactly 120120 items: t+b=120t + b = 120, so t=120−bt = 120 - b.
    • The tacos bring in 4(120−b)4(120 - b) dollars. That is 480−4b480 - 4b.
    • The burritos bring in 7b7b dollars.
    • Add them: −4b+7b=3b-4b + 7b = 3b, so the sales are 480+3b480 + 3b dollars.
    • The sales must be at least 600600: 480+3b≥600480 + 3b \ge 600.
    • Subtract 480480 from both sides: 3b≥1203b \ge 120.
    • Divide both sides by 33: b≥40b \ge 40. The least number of burritos is 4040.
    • Check: 7⋅40=2807 \cdot 40 = 280 and 4⋅80=3204 \cdot 80 = 320, and 280+320=600280 + 320 = 600 dollars.
The correct answer is 4040.

The same example in Desmos

  1. new graph

    The item limit, with xx for tacos and yy for burritos.

  2. The sales goal.

  3. zoom out until the lines cross, then click the crossing

    (80,40)(80, 40): 8080 tacos and 4040 burritos. At this corner the sales are exactly $600\$600.

Type
Desmos expression list: x+y=120; 4x+7y=600
Look
Desmos graph of x+y=120; 4x+7y=600, with (80, 40) labelled

Made with Desmos

Open Desmos

Your turn

A workshop runs two machines for at most 2020 hours a week in total. Machine A makes 1515 parts an hour and machine B makes 1010 parts an hour. The workshop needs at least 240240 parts a week. In a week when the machines run for exactly 2020 hours in total, what is the least number of hours that machine A must run?

Show the answerHide the answer

Answer

The correct answer is 88.

  1. Step 1: Identify the two variables

    • Let xx be the hours that machine A runs.
    • Let yy be the hours that machine B runs.

  2. Step 2: Determine the inequality for each constraint

    • Hours: ‘at most 2020’ means x+y≤20x + y \le 20.
    • Parts: ‘at least 240240’ means 15x+10y≥24015x + 10y \ge 240.

  3. Step 3: Solve the two boundaries at the corner

    • The machines run for exactly 2020 hours: x+y=20x + y = 20, so y=20−xy = 20 - x.
    • Machine A makes 15x15x parts.
    • Machine B makes 10(20−x)10(20 - x) parts. That is 200−10x200 - 10x.
    • Add them: 15x−10x=5x15x - 10x = 5x, so the total is 5x+2005x + 200 parts.
    • The total must be at least 240240: 5x+200≥2405x + 200 \ge 240.
    • Subtract 200200 from both sides: 5x≥405x \ge 40.
    • Divide both sides by 55: x≥8x \ge 8. Machine A must run for at least 88 hours.

Traps

  • Turning the symbol the wrong way. ‘At most 120120’ is ≤120\le 120, and ‘at least $600\$600’ is ≥600\ge 600. Writing t+b≥120t + b \ge 120 says the truck makes 120120 items or more, which it cannot.
  • Swapping the prices. 7t+4b≥6007t + 4b \ge 600 prices a taco at $7\$7. At the corner the sales become 840−3b840 - 3b dollars, which gives b≤80b \le 80, a wrong answer.
Practice this lesson24 questions

When two models are equal

Key points

  • Two plans, or two things that change, each have their own expression.
  • ‘The same cost’, ‘the same amount’ and ‘catches up’ all mean: set the two expressions equal.
  • Solving gives the input: when it happens, or how many.
  • If the question asks for the shared amount, plug the input back in.
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See it solved

Example

Tank A holds 4040 gallons of water and gains 66 gallons each minute. Tank B holds 100100 gallons and gains 22 gallons each minute. At some moment the two tanks hold the same amount of water. How many gallons does each tank hold at that moment?

Worked by hand

The plan

  1. Equate the two expressions
  2. Determine the value of tt
  3. Plug the input into the function

Once the two expressions are written, Desmos is quicker: graph both and click the crossing. It shows the input and the shared amount together.

  1. Step 1: Equate the two expressions

    • Let tt be the number of minutes.
    • Tank A holds 40+6t40 + 6t gallons. Tank B holds 100+2t100 + 2t gallons.
    • The same amount: 40+6t40 + 6t =100+2t= 100 + 2t.

  2. Step 2: Determine the value of tt

    • Subtract 2t2t from both sides: 40+4t=10040 + 4t = 100.
    • Subtract 4040 from both sides: 4t4t =100−40= 100 - 40 =60= 60.
    • t=604=15t = \dfrac{60}{4} = 15.

  3. Step 3: Plug the input into the function

    • Tank A: 40+6⋅1540 + 6 \cdot 15.
    • 6⋅15=906 \cdot 15 = 90, so 40+90=13040 + 90 = 130.
    • Check with tank B: 100+2⋅15100 + 2 \cdot 15 =100+30= 100 + 30 =130= 130. It matches.
The correct answer is 130130.

The same example in Desmos

  1. new graph

    Tank A, with xx for the minutes.

  2. Tank B.

  3. zoom out until the lines cross, then click the crossing

    (15,130)(15, 130): after 1515 minutes, each tank holds 130130 gallons.

Type
Desmos expression list: y=40+6x; y=100+2x
Look
Desmos graph of y=40+6x; y=100+2x, with (15, 130) labelled

Made with Desmos

Open Desmos

Your turn

A ride with company P costs $4\$4 plus $2\$2 per mile. A ride with company Q costs $10\$10 plus $1.50\$1.50 per mile. For how many miles do the two rides cost the same?

Show the answerHide the answer

Answer

The correct answer is 1212.

  1. Step 1: Equate the two expressions

    • Let mm be the number of miles.
    • Company P costs 4+2m4 + 2m dollars. Company Q costs 10+1.5m10 + 1.5m dollars.
    • The same cost: 4+2m4 + 2m =10+1.5m= 10 + 1.5m.

  2. Step 2: Determine the value of mm

    • Subtract 1.5m1.5m from both sides: 4+0.5m=104 + 0.5m = 10.
    • Subtract 44 from both sides: 0.5m0.5m =10−4= 10 - 4 =6= 6.
    • m=60.5=12m = \dfrac{6}{0.5} = 12.
    • Check: 4+2⋅124 + 2 \cdot 12 =4+24= 4 + 24 =28= 28 and 10+1.5⋅1210 + 1.5 \cdot 12 =10+18= 10 + 18 =28= 28. It matches.

Traps

  • Stopping at the input. t=15t = 15 is when the tanks match. The question asks how many gallons, so plug 1515 back in to get 130130. 1515 is a wrong answer.
  • Mixing up the start and the rate. 4040 and 100100 are the starting amounts; 66 and 22 go with tt. Writing 6+40t6 + 40t =2+100t= 2 + 100t gives 4=60t4 = 60t, so t=460t = \dfrac{4}{60}, a wrong answer.
Practice this lesson20 questions
Typing tips for Desmos
  • In the keys lines, → means the right-arrow key.
  • Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
  • Desmos solves equations in x (and y) only. An equation in t, such as 3t + 7 = 22, gives an error until you retype it with x.
  • Type y1 and Desmos shows y₁: a digit typed straight after a letter becomes a subscript. If you type the underscore yourself (y_1), press → after the digit: a ~ typed next lands inside the subscript (+, − and = leave it by themselves).
  • The ~ key is Shift + ` on most keyboards (the key left of 1). Use ~, not =: ~ asks Desmos to find the unknown letters; = only draws or defines.
  • A letter that already has a value, such as a slider m = 1 left over from earlier, is held fixed by a ~ fit, with no warning. Start a new graph, or delete old sliders, before each question.
  • Start a new graph for each example. Rows left from an earlier example clash with the new ones: a second f(x)f(x), or a second fit for mm and bb, is an error.
  • Type y1 and Desmos shows y1y_1: a digit typed straight after a letter becomes a subscript.
  • The tilde, ~, is Shift and the key left of 1 on most keyboards. Use ~, not =: ~ finds the letters that make the line fit; = only draws or defines.
  • Desmos works in xx and yy. Type any other letter, such as hh or tt, as xx.
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