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 solve by substitution
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Algebra
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Algebra
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Systems of Linear Equations
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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
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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 9-12
Access to these problems requires a Membership.
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The ABC's of Exponents
- Steve Risberg
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Algebra, difficulty level 3. Solve a system of equations involving exponents.
... more>>
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Adrienne's Axel
- Leslie Hogben
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Algebra, difficulty level 2. Adrienne, an algebraically inclined figure skater, figures an interesting problem for us.
... more>>
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The Ages of Sigma
- Terry Trotter
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Algebra, difficulty level 1. Find the ages of the three artists who make up the latest singing sensation: Sigma.
... more>>
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Bowling Buddies
- Steve Risberg
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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>>
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The Box Supper
- Terry Trotter
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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>>
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Buried Treasure
- Steve Risberg
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Algebra, difficulty level 3. The pirate Algebeard has drawn an unusual treasure map. Can you
determine where his treasure is buried?
... more>>
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Choosing Charity
- Steve Risberg
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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.
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Crossed Wires
- Steve Risberg
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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>>
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The Custom of Customs
- Ethel Breuche
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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>>
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Eduardo's Equations
- Steve Risberg Annie Fetter
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Algebra, difficulty level 1. Find all possible integer solutions to a system of four equations in
two variables.
... more>>
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Escalation
- Melissa Kennedy
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Algebra, difficulty level 2. A rate, time, distance problem that involves finding the number of steps on an escalator.
... more>>
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Flatland Soldiers
- Terry Trotter
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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.
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Four Men and Some Money
- Daboll
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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>>
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Grandpa Jones' Age
- Terry Trotter
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Algebra, difficulty level 2. Little George asks how old his grandpa is, and needs your help to find out the answer.
... more>>
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The Great Doughnut Caper
- Terry Trotter
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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>>
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Gumballs Galore!
- Terry Trotter
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Algebra, difficulty level 2. Two famous characters from literature go on a gumball shopping spree.
... more>>
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Highway Miles
- Terry Trotter
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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>>
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Hole in One!
- Steve Risberg
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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>>
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Holiday Ornaments
- Claire Mead
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Math Fundamentals, difficulty level 3. How many holiday ornaments did each girl's mother make?
... more>>
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Hoop Heroes
- Steve Risberg
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Algebra, difficulty level 3. Use inequalities to determine how many points Kylie and Shaniqua each
scored in the championship basketball game.
... more>>
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Humphrey's Great Escape
- Bill Graham
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Algebra, difficulty level 3. Calculate Humphrey's running speed as he tries to outrun a train across a trestle.
... more>>
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Jimmy's Hits
- Terry Trotter & Annie Fetter
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Algebra, difficulty level 2. Given information about how many doubles, homers, and singles, figure out how many total hits Jimmy had during his record-breaking season.
... more>>
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Johnny's Orchards
- Terry Trotter
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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>>
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Let It Snow!
- Steve Risberg
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Algebra, difficulty level 3. Use the given statements to determine how many clear days there were
in a recent span of snowy days.
... more>>
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Lieutenant Stone's Troops
- Terry Trotter
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Algebra, difficulty level 2. Find the number of soldiers, performing in a formal military ceremony, who were marching in various rectangular arrangements.
... more>>
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Lost in Mathland
- Steve Risberg
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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>>
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Math Club Mystery
- Steve Risberg
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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.
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The Mensa Toy Store
- Terry Trotter
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Algebra, difficulty level 2. The prices are strange in this store. Can you decipher them?
... more>>
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Milly Miffin's Muffins
- J. A. H. Hunter
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Algebra, difficulty level 2. Milly Miffin et al. are muffin makers extraordinaires, and they make muffins mathematically, too. The question is, how many?
... more>>
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The Mystery Quadrilateral
- Annie Fetter & Dr. Math
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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>>
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My Three Dogs
- Terry Trotter
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Algebra, difficulty level 2. Given information about the weights of three pairs of dogs, figure out the dogs' individual weights.
... more>>
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Parakeets on Parade
- Terry Trotter
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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>>
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Peanuts and Cashews
- Terry Trotter
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Algebra, difficulty level 1. Find the proper mixture of nuts that have a certain price rate to be sold in the nut store.
... more>>
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The Professors' Ages
- Terry Trotter
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Algebra, difficulty level 3. Find the professors' ages to complete the requested calculation.
... more>>
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Ration Ratios
- Steve Risberg
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Algebra, difficulty level 3. Determine the ratio of vegetarian meals to vegetarians at the luncheon.
... more>>
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Shanell's Holiday Treats
- Terry Trotter
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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>>
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Splash!
- Steve Risberg
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Algebra, difficulty level 1. How many throws did it take each student to sink Mr. Rashanul in the
dunking booth at the carnival?
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Splitting a Square
- Terry Trotter
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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?
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The Sprats
- Terry Trotter & Lisa Lavelle
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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>>
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Studying the Standings
- Steve Risberg
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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>>
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Sums, Quotients, and Remainders
- Terry Trotter
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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.
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Take Me Out to the Ballgame
- Annie Fetter
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Algebra, difficulty level 4. Determine how many innings a pitcher has thrown as his Earned Run Average changes during a game.
... more>>
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Teeter Troop
- Steve Risberg
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Algebra, difficulty level 3. Four animals are balancing a seesaw. Can you determine how much each
one weighs?
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The Coordinate Carousel
- Steve Risberg
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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>>
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The Three Jelly Bean Jars
- Terry Trotter
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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>>
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Triangle Probability
- Annie Fetter
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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 x-coordinate of the selected point is less than the y-coordinate?
... more>>
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Tribonacci
- Terry Trotter
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Algebra, difficulty level 3. Find the second and third terms of a "Tribonacci" sequence.
... more>>
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Two for Two
- Annie Fetter
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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>>
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Two for Two More
- Annie Fetter
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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>>
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The Unhappy Carpenter
- Ethel Breuche
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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>>
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The Variable Rectangle
- Terry Trotter and Friends
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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>>
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Vexing Variables
- Steve Risberg
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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>>
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Walter's Weird Wages
- Terry Trotter
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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>>
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Weighing Kittens
- Terry Trotter
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Algebra, difficulty level 2. Given some facts, figure out how many ounces each of seven kittens weighs.
... more>>
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What's the Difference?
- Steve Risberg
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Algebra, difficulty level 3. Does multiplying a number by four always give you a greater number
than you started with?
... more>>
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What's what?
- Terry Trotter (author unknown)
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Algebra, difficulty level 2. A poetical puzzle to ponder about: "What is WHAT?"
... more>>
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Winter Gold
- Terry Trotter
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Algebra, difficulty level 2. Given some clues, figure out how many gold medals were won by each country.
... more>>
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X Marks the Spot
- Terry Trotter
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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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