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Systems of Linear Equations

These problems can be solved by using two or more variables and writing a system of equations. Some students may choose to solve some of the problems by using a single
variable and equation. System solution methods involve the use of substitution or combination techniques.
Some of these problems are also in the following subcategories:
 Related Resources

Interactive resources from our Math Tools project:
Algebra: Linear Systems
The closest match in our Ask Dr. Math archives:
High School: Linear Equations
NCTM Standards:
Algebra Standard for Grades 912
Access to these problems requires a Membership.

The ABC's of Exponents
 Steve Risberg

Algebra, difficulty level 3. Solve a system of equations involving exponents.
... more>>

Adrienne's Axel
 Leslie Hogben

Algebra, difficulty level 2. Adrienne, an algebraically inclined figure skater, figures an interesting problem for us.
... more>>

The Ages of Sigma
 Terry Trotter

Algebra, difficulty level 1. Find the ages of the three artists who make up the latest singing sensation: Sigma.
... more>>

Bowling Buddies
 Steve Risberg

Algebra, difficulty level 2. Four friends go bowling and have a small contest. From the clues
given about their scores, can you determine who won the contest?
... more>>

The Box Supper
 Terry Trotter

Algebra, difficulty level 3. Find the amount of money that Judy Ann's "beau" paid at the box supper auction for her box of delicious food.
... more>>

Buried Treasure
 Steve Risberg

Algebra, difficulty level 3. The pirate Algebeard has drawn an unusual treasure map. Can you
determine where his treasure is buried?
... more>>

Choosing Charity
 Steve Risberg

Algebra, difficulty level 2. A company donates to a relief fund by giving a certain amount plus an
additional amount for each employee who decides to donate. Given two
data points, find the linear equation that defines the company's part
of the total donation.
... more>>

Crossed Wires
 Steve Risberg

Algebra, difficulty level 3. Guy wires support two poles of different heights. Find the height of
their intersection and explore what happens as the distance between
the poles changes.
... more>>

The Custom of Customs
 Ethel Breuche

Algebra, difficulty level 2. Two wine merchants enter Paris with their merchandise. They need to pay custom duties. Each pays with francs and with some casks of wine. What is each cask of wine worth and what is the duty on each cask?
... more>>

Eduardo's Equations
 Steve Risberg Annie Fetter

Algebra, difficulty level 1. Find all possible integer solutions to a system of four equations in
two variables.
... more>>

Escalation
 Melissa Kennedy

Algebra, difficulty level 2. A rate, time, distance problem that involves finding the number of steps on an escalator.
... more>>

Flatland Soldiers
 Terry Trotter

Algebra, difficulty level 1. Find the size of the vertex angle of a soldier in the world of Flatland, a famous book by Edwin A. Abbott.
... more>>

Four Men and Some Money
 Daboll

Algebra, difficulty level 1. Four men (A, B, C, & D) are splitting up a quantity of money ($100), and need your help to do it.
... more>>

Grandpa Jones' Age
 Terry Trotter

Algebra, difficulty level 2. Little George asks how old his grandpa is, and needs your help to find out the answer.
... more>>

The Great Doughnut Caper
 Terry Trotter

Algebra, difficulty level 2. After a number of doughnuts have been switched in their respective boxes, you are to discover the original state of things before the change was made.
... more>>

Gumballs Galore!
 Terry Trotter

Algebra, difficulty level 2. Two famous characters from literature go on a gumball shopping spree.
... more>>

Highway Miles
 Terry Trotter

Algebra, difficulty level 2. Find the number of miles a person travels on a highway, given the number of gallons of gas and the consumption rate.
... more>>

Hole in One!
 Steve Risberg

Algebra, difficulty level 4. Given the dimensions of a miniature golf hole, use slopes to determine
how Tiger needs to hit his shot to make a hole in one.
... more>>

Holiday Ornaments
 Claire Mead

Math Fundamentals, difficulty level 3. How many holiday ornaments did each girl's mother make?
... more>>

Hoop Heroes
 Steve Risberg

Algebra, difficulty level 3. Use inequalities to determine how many points Kylie and Shaniqua each
scored in the championship basketball game.
... more>>

Humphrey's Great Escape
 Bill Graham

Algebra, difficulty level 3. Calculate Humphrey's running speed as he tries to outrun a train across a trestle.
... more>>

Jimmy's Hits
 Terry Trotter & Annie Fetter

Algebra, difficulty level 2. Given information about how many doubles, homers, and singles, figure out how many total hits Jimmy had during his recordbreaking season.
... more>>

Johnny's Orchards
 Terry Trotter

Algebra, difficulty level 3. Johnny Appleseed plants three rectangular orchards of various shapes.
If each orchard holds the same number of seeds, how many seeds did he
plant?
... more>>

Let It Snow!
 Steve Risberg

Algebra, difficulty level 3. Use the given statements to determine how many clear days there were
in a recent span of snowy days.
... more>>

Lieutenant Stone's Troops
 Terry Trotter

Algebra, difficulty level 2. Find the number of soldiers, performing in a formal military ceremony, who were marching in various rectangular arrangements.
... more>>

Lost in Mathland
 Steve Risberg

Algebra, difficulty level 2. Zach and Zeke get separated while hiking in Mathland, where roads
are named with equations. Zeke is at the intersection of three roads
 can you help Zach determine where he is?
... more>>

Math Club Mystery
 Steve Risberg

Algebra, difficulty level 2. Given information about the total number of people, ticket prices, and
money spent, determine how many students, teachers, and parents went
on the Math Club field trip.
... more>>

The Mensa Toy Store
 Terry Trotter

Algebra, difficulty level 2. The prices are strange in this store. Can you decipher them?
... more>>

Milly Miffin's Muffins
 J. A. H. Hunter

Algebra, difficulty level 2. Milly Miffin et al. are muffin makers extraordinaires, and they make muffins mathematically, too. The question is, how many?
... more>>

The Mystery Quadrilateral
 Annie Fetter & Dr. Math

Algebra, difficulty level 2. Three consecutive sides of a quadrilateral are 2, 6, 9, and x. If
the
diagonals are perpendicular, what's x?
... more>>

My Three Dogs
 Terry Trotter

Algebra, difficulty level 2. Given information about the weights of three pairs of dogs, figure out the dogs' individual weights.
... more>>

Parakeets on Parade
 Terry Trotter

Algebra, difficulty level 2. A flock of parakeets flies a round trip to the feeding ground and
back. Given the total time and distance of the round trip, and the
speed of the wind in each direction, find the flying speed of the
parakeets.
... more>>

Peanuts and Cashews
 Terry Trotter

Algebra, difficulty level 1. Find the proper mixture of nuts that have a certain price rate to be sold in the nut store.
... more>>

The Professors' Ages
 Terry Trotter

Algebra, difficulty level 3. Find the professors' ages to complete the requested calculation.
... more>>

Ration Ratios
 Steve Risberg

Algebra, difficulty level 3. Determine the ratio of vegetarian meals to vegetarians at the luncheon.
... more>>

Shanell's Holiday Treats
 Terry Trotter

Algebra, difficulty level 3. Given some data about purchasing fancy nuts for cooking, find the total cost of buying one pound of each type.
... more>>

Splash!
 Steve Risberg

Algebra, difficulty level 1. How many throws did it take each student to sink Mr. Rashanul in the
dunking booth at the carnival?
... more>>

Splitting a Square
 Terry Trotter

Algebra, difficulty level 3. A square is drawn on a coordinate plane with vertices P(1, 0), Q(2, 0), R(2, 1) and S(1, 1). A line is drawn from the origin that cuts the square into two sections whose areas are in a ratio of 2:1. What's the equation of the "cutting" line?
... more>>

The Sprats
 Terry Trotter & Lisa Lavelle

Algebra, difficulty level 2. The Sprats go out to dinner three times, and at each meal they order a different combination of two of their three favorite dishes. Given the bill for each meal, determine how much the Sprats' favorites cost individually.
... more>>

Studying the Standings
 Steve Risberg

Algebra, difficulty level 3. Write and solve a system of equations to determine how many points the
Potters have in the World Tiddlywinks League standings.
... more>>

Sums, Quotients, and Remainders
 Terry Trotter

Algebra, difficulty level 3. Given the sum of two numbers and information about the quotient and
remainder when they are divided, find the larger number, then
solve the same sort of problem in a general case.
... more>>

Take Me Out to the Ballgame
 Annie Fetter

Algebra, difficulty level 4. Determine how many innings a pitcher has thrown as his Earned Run Average changes during a game.
... more>>

Teeter Troop
 Steve Risberg

Algebra, difficulty level 3. Four animals are balancing a seesaw. Can you determine how much each
one weighs?
... more>>

The Coordinate Carousel
 Steve Risberg

Algebra, difficulty level 3. Mrs. Nomial is watching her daughters on the carousel and notices
something interesting. Determine how far away from the edge of the
carousel she is standing.
... more>>

The Three Jelly Bean Jars
 Terry Trotter

Algebra, difficulty level 2. Three brothers buy jelly beans. The jelly beans are in jars that are three different sizes. How many jelly beans does each brother buy?
... more>>

Triangle Probability
 Annie Fetter

Algebra, difficulty level 3. A point is randomly selected from within a triangle with vertices at (0,0), (0,4) and (6,0). What is the probability that the xcoordinate of the selected point is less than the ycoordinate?
... more>>

Tribonacci
 Terry Trotter

Algebra, difficulty level 3. Find the second and third terms of a "Tribonacci" sequence.
... more>>

Two for Two
 Annie Fetter

Algebra, difficulty level 3. Two numbers have a ratio of 5:3. The larger number is 36 more than half the difference of the two numbers. What are the two numbers?
... more>>

Two for Two More
 Annie Fetter

Algebra, difficulty level 3. Two numbers have a ratio of 4:3. The greater number is 60 more than 8 times their difference. What could the two numbers be?
... more>>

The Unhappy Carpenter
 Ethel Breuche

Algebra, difficulty level 1. A carpenter agrees to be paid a certain amount of money when she works, and to repay money on the days she doesn't work. Find the number of days she worked if she earned $77.00.
... more>>

The Variable Rectangle
 Terry Trotter and Friends

Algebra, difficulty level 1. Increase a rectangle's width and decrease its length each by 1 and the area is decreased by 4. But decrease its width by 1 and increase its length by 2, and the area is increased by 16. What's the area of the original rectangle?
... more>>

Vexing Variables
 Steve Risberg

Algebra, difficulty level 3. Use two algebraic statements involving three variables to find the
numeric value of the ratio of two of those variables.
... more>>

Walter's Weird Wages
 Terry Trotter

Algebra, difficulty level 2. Walter's eccentric parents give their son his housework wages in a strange way. Can you determine how much he receives?
... more>>

Weighing Kittens
 Terry Trotter

Algebra, difficulty level 2. Given some facts, figure out how many ounces each of seven kittens weighs.
... more>>

What's the Difference?
 Steve Risberg

Algebra, difficulty level 3. Does multiplying a number by four always give you a greater number
than you started with?
... more>>

What's what?
 Terry Trotter (author unknown)

Algebra, difficulty level 2. A poetical puzzle to ponder about: "What is WHAT?"
... more>>

Winter Gold
 Terry Trotter

Algebra, difficulty level 2. Given some clues, figure out how many gold medals were won by each country.
... more>>

X Marks the Spot
 Terry Trotter

Algebra, difficulty level 3. A Spanish pirate buries his treasure on an island in a very mathematical manner. Can you locate the "X" spot?
... more>>
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