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What's an Example of a Word Problem That Has a System of Linear Equations with Infinite Solutions?

Batman is trying to find the location of the Riddler using coordinates on a map. The Riddler leaves him a clue, telling him that if you multiply the first coordinate by 2 and subtract the second coordinate, he’ll get 2. He also tells him that if he doubles the second coordinate and subtracts that from four times the first coordinate, he’ll get 4. Can the Riddler be found?

Summary

  1. 'x' is a variable that represents the first coordinate
  2. 'y' is a variable that represents the second coordinate
  3. Distribute in the 2 by multiplying by each term in the first equation
  4. Multiplying the first equation by 2 gives us the second equation, 4x-2y=4
  5. This means there are an infinite number of solutions to this system

Notes

    1. We'll use the variable 'x' to represent the first coordinate
    2. We'll use the variable 'y' to represent the second coordinate
    1. For the first equation, we multiply the first coordinate, 'x', by 2 to get 2x
    2. Then we subtract the second coordinate, 'y', and get 2
    3. So the first equation is 2x-y=2
    4. For the second equation, we double the second coordinate, 'y', to get 2y
    5. We want to subtract this from 4 times the first coordinate, which is would be 4x
    6. When we subtract these we get 4, so the second equation is 4x-2y=4
    1. We can see that each term in the second equation is 2 times each term in the first equation
    2. So if we multiply the first equation by 2, we should get the second equation
    3. Multiply 2•2x to get 4x
    4. Multiply 2•(-y) to get -2y
    5. Multiply 2•2 to get 4
    6. This gives us 4x-2y=4, which was our second equation!
    1. Consistent and dependent means that our system has an infinite number of solutions
    2. The same (x,y) ordered pairs are solutions to each equation.
    1. There are an infinite number of (x,y) coordinate pairs where the Riddler could be
    2. So he could be anywhere!