Write a System of Equations

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Key Ideas Easy Medium Hard Super Challenge

What to Remember

Name both variables first

Say what each letter stands for, with units. Two unknowns need two letters.

Two facts, two equations

Each sentence that relates the two quantities becomes one equation. A system is the pair of equations together.

Count and value

A common pair is (number of items) and (total cost or total amount). Keep the units the same in each equation.

Check that you used both facts

If one equation uses only one letter, you may have combined the facts too soon. Write both relationships first.

Level 1

Easy

Read the story. Name the letters, then write two equations. You do not have to solve yet.

Example

Popcorn costs \$4 and drinks cost \$2. A group buys 7 items and spends \$20. Write a system for the number of popcorns p and drinks d.

Let p = the number of popcorns.

Let d = the number of drinks.

Count: p + d = 7

Cost: 4p + 2d = 20

Answer: p + d = 74p + 2d = 20

Adult tickets cost \$12 and child tickets cost \$7. A family buys 9 tickets and spends \$88. Write a system for adults a and children c.

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Let a = the number of adult tickets.

Let c = the number of child tickets.

a + c = 9

12a + 7c = 88

Answer: a + c = 912a + 7c = 88

Apples cost \$2 each and oranges cost \$3 each. You buy 10 pieces of fruit and spend \$24. Write a system for apples a and oranges o.

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Let a = the number of apples.

Let o = the number of oranges.

a + o = 10

2a + 3o = 24

Answer: a + o = 102a + 3o = 24

Two numbers have sum 18 and difference 4. Write a system for the larger number ℓ and the smaller number s.

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Let ℓ = the larger number.

Let s = the smaller number.

ℓ + s = 18

ℓ − s = 4

Answer: ℓ + s = 18ℓ − s = 4

Shirts cost \$15 and hats cost \$8. A shopper buys 6 items and spends \$76. Write a system for shirts s and hats h.

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Let s = the number of shirts.

Let h = the number of hats.

s + h = 6

15s + 8h = 76

Answer: s + h = 615s + 8h = 76

Maya walks at 3 miles per hour and runs at 6 miles per hour. She exercises for 5 hours and covers 21 miles. Write a system for walking hours w and running hours r.

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Let w = hours walking.

Let r = hours running.

w + r = 5

3w + 6r = 21

Answer: w + r = 53w + 6r = 21

Level 2

Medium

The two facts may be two different combinations, or a total plus a comparison.

Example

Three bags of chips and two sodas cost \$13. One bag of chips and four sodas cost \$11. Write a system for the price of a bag of chips c and a soda s.

Let c = the price of one bag of chips in dollars.

Let s = the price of one soda in dollars.

3c + 2s = 13

c + 4s = 11

Answer: 3c + 2s = 13c + 4s = 11

A café sells coffee for \$4 and tea for \$3. They sold twice as many coffees as teas and took in \$33. Write a system for coffees c and teas t.

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Let c = the number of coffees.

Let t = the number of teas.

c = 2t

4c + 3t = 33

Answer: c = 2t4c + 3t = 33

The length of a rectangle is 5 more than the width. The perimeter is 38. Write a system for length ℓ and width w.

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Let ℓ = the length.

Let w = the width.

ℓ = w + 5

2ℓ + 2w = 38

Answer: ℓ = w + 52ℓ + 2w = 38

A jar has nickels and dimes. There are 18 coins worth \$1.50. Write a system for nickels n and dimes d. Use cents for the value equation.

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Let n = the number of nickels.

Let d = the number of dimes.

n + d = 18

5n + 10d = 150

Answer: n + d = 185n + 10d = 150

Four weekday visits and two weekend visits cost \$52. Three weekday visits and five weekend visits cost \$61. Write a system for the weekday price w and the weekend price e.

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Let w = the weekday price in dollars.

Let e = the weekend price in dollars.

4w + 2e = 52

3w + 5e = 61

Answer: 4w + 2e = 523w + 5e = 61

A boat travels 24 miles downstream in 2 hours and 18 miles upstream in 3 hours. Let b be the boat’s speed in still water and c the current, both in miles per hour. Write a system.

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Downstream speed is b + c. Upstream speed is b − c.

Distance = (rate)(time)

2(b + c) = 24

3(b − c) = 18

Or, after dividing: b + c = 12 and b − c = 6

Answer: b + c = 12b − c = 6

Level 3

Hard

Percents, two investments, and mixed units still become two linear equations.

Example

A chemist mixes a 15% acid solution with a 40% acid solution to make 20 liters of 25% acid. Write a system for liters of 15% solution t and liters of 40% solution h.

Let t = liters of 15% solution.

Let h = liters of 40% solution.

t + h = 20

0.15t + 0.40h = 0.25(20)

0.15t + 0.40h = 5

Answer: t + h = 200.15t + 0.40h = 5

You invest \$5,000 in two accounts. One earns 4% and the other earns 7%. The total interest for the year is \$275. Write a system for the amount x at 4% and the amount y at 7%.

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Let x = dollars in the 4% account.

Let y = dollars in the 7% account.

x + y = 5000

0.04x + 0.07y = 275

Answer: x + y = 50000.04x + 0.07y = 275

Nuts that cost \$6 per pound are mixed with nuts that cost \$9 per pound to make 12 pounds that cost \$7.50 per pound. Write a system for pounds of \$6 nuts a and pounds of \$9 nuts b.

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Let a = pounds of \$6 nuts.

Let b = pounds of \$9 nuts.

a + b = 12

6a + 9b = 7.50(12)

6a + 9b = 90

Answer: a + b = 126a + 9b = 90

Two large pizzas and three small pizzas cost \$31. Four large pizzas and one small pizza cost \$33. Write a system for the large price L and the small price S.

Show Answer

Let L = the price of a large pizza.

Let S = the price of a small pizza.

2L + 3S = 31

4L + S = 33

Answer: 2L + 3S = 314L + S = 33

A plane flies with a 30 mph tailwind and covers 900 miles in the same time it covers 750 miles against that wind. Let p be the plane’s speed in still air and t the time in hours. Write a system.

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With the wind, speed is p + 30.

Against the wind, speed is p − 30.

(p + 30)t = 900

(p − 30)t = 750

Answer: (p + 30)t = 900(p − 30)t = 750

Three times a first number plus a second number is 28. The first number plus four times the second is 31. Write a system for the first number x and the second number y.

Show Answer

Let x = the first number.

Let y = the second number.

3x + y = 28

x + 4y = 31

Answer: 3x + y = 28x + 4y = 31

Super Challenge

Adult tickets are \$11 and student tickets are \$6. A school sells 40 tickets and collects \$355. Write a system, then explain what a negative student count would mean if the algebra produced one.

Show Answer

Let a = adult tickets and s = student tickets.

a + s = 40

11a + 6s = 355

A negative s would mean the ticket counts cannot both be true with these prices. In the story, ticket counts cannot be negative, so that pair would not be a real sale.

Answer: a + s = 4011a + 6s = 355

A 10% solution and a 50% solution are mixed to make 8 liters of k% solution. Write a system in t, h, and k. What extra fact do you need before you can find t and h?

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Let t = liters of 10% and h = liters of 50%.

t + h = 8

0.10t + 0.50h = k100(8)

You still have three letters. You need a value for k (or one of the amounts) before the system has a unique pair (t, h).

Answer: t + h = 80.10t + 0.50h = 0.08k

Two phone plans. Plan A is \$20 plus \$5 per gigabyte. Plan B is \$8 per gigabyte with no starting fee. Write a system that finds the gigabytes g and the matching cost C where the plans cost the same.

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Let g = gigabytes and C = the common cost in dollars.

C = 20 + 5g

C = 8g

The solution is the break-even pair (g, C).

Answer: C = 20 + 5gC = 8g

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