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How Do You Use a System of Linear Equations to Find Coordinates on a Map?

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, you get 2. He also tells him that if you add the second coordinate to twice the first coordinate, you get 6. What are the map coordinates of the Riddler?

Summary

  1. 'x' and 'y' are variables that represent the values of the Riddler's coordinates
  2. We need to make equations to determine the value of each variable, or each coordinate
  3. By adding -2x-y=-6 to 2x-y=2, we'll eliminate x and be able to solve for y
  4. Plug y=2 into 2x-y=2 in order to solve for 'x'
  5. Plug y=2 into 2x+y=6 to check the solution for 'x'
  6. We can write our solution as an ordered pair (x,y)=(2,2)

Notes

    1. Variables are letters that represent unknown values
    2. In this problem, we need to find the first and second coordinates to locate the Riddler
    1. We can use the problem statement to write equations containing 'x' and 'y'
    2. Multiplying the first coordinate, 'x', by 2 gives us 2x
    3. Subtract the second coordinate, 'y', from 2x to get 2
    4. That means our first equation is 2x-y=2
    5. If we add the second coordinate, 'y', to 2 times the first coordinate, 'x', that equals 6
    6. So our second equation is 2x+y=6
    1. To solve for y, we can combine the equations in a special way and eliminate 'x'
    2. There are many ways to solve a system of equations!
    3. Check out the links below to see some other ways to solve this system:
    1. To solve for y, we need to eliminate x
    2. Multiply 2x+y=6 by -1 so that the 'x' term will cancel out when we add
    3. Multiplying an equation by -1 flips all of the signs!
    1. Adding the resulting equations cancels out 2x and leaves -2y by itself
    2. To isolate 'y' we need to divide both sides by -2
    3. The -2's cancel on the left
    4. On the right, (-4)/(-2)=2
    1. To solve for x, we'll plug y into our first equation
    1. To solve for x, we'll plug y into our first equation
    2. Once y is plugged in, we have to get x by itself
    3. Use the Subtraction Property of Equality to subtract 2 from both sides and isolate 2x
    4. To isolate 'x' from '2x' we need to divide both sides by 2
    5. The 2's cancel on the left
    6. On the right, 4/2=2
    1. We can write our solution as an ordered pair (x,y)=(2,2)
    2. Since we have one solution to this system of equations, our system is 'consistent' and 'independent'