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Two Methods:Using the Method of Addition/EliminationUsing the Method of Substitution

Simultaneous equations are a pair of equations with two unknown values. Are you trying to solve simultaneous equations algebraically? The basic process is easy to master, with some practice. Read on to learn how to solve these equations, and you'll be well on your way towards tackling more complicated examples.

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Method 1 of 2: Using the Method of Addition/Elimination

  1. 1
    Examine your pair of equations. You've most likely been presented with two equations containing x and y. For example:
    • 2x + y = 11
    • 3x - 2y = 6
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  2. 2
    Start by making the y coefficients the same. You need them to be equal to one another, in the different equations, so that you can cancel them out. Then you'll be able to solve for just x.
    • If they are already the same, move on.
    • If they are not, then you can do this by multiplying both the equations by the LCM (lowest common multiple) of both the y coefficient values. For example:
      • With the examples provided, your lowest common multiple is 2.
      • 2x + y = 11 multiplied by 2 becomes: 4x + 2y = 22
      • 3x - 2y = 6 doesn't have to change, because it's already got a 2 as the y coefficient.
    • Remember, you can multiple the equation by any number you like without making it inaccurate, as long as you multiply both sides.
  3. 3
    Add or subtract the equations vertically. Think about whether you need to add both the equations together or take them away. You want the y values to cancel one another out. If in front of both y values there is a minus sign then you add the equations, but if one is an addition and the other is a subtraction, then subtract the equations.
    • For example:
      • 4x + 2y = 22
      • + 3x - 2y = 6
      • ------------
      • 7x = 28
    • Remember: If the signs are the same, subtract; if they're different, add.
  4. 4
    Solve for x. To do this in the example, divide the solution of the equation (28) by the x coefficient, or number before the x, (7). In this example, x = 4.
  5. 5
    Substitute x in the second equation. Use algebra to work out y.
    • In this case, 4(4) + 2y = 22, so 16 + 2y = 22 and 2y = 6. This tells you that y=3.
  6. 6
    Check it in the first equation. If you found the right solution, the x and y values will make both equations true.
    • In this example, 2x + y = 11 comes out to 2(4) + 3 = 11, which is true, and 3x - 2y = 6 comes out to 3(4) - 2(3) = 6, which is also true, so your results for x and y are accurate. Well done!

Method 2 of 2: Using the Method of Substitution

  1. 1
    Examine your pair of equations. You've most likely been presented with two equations containing x and y. For example:
    • 2x + y = 11
    • 3x - 2y = 6
  2. 2
    Solve one equation for y. You need to isolate a variable on one side of your equation.
    • In this case, the easiest one to use is 2x + y = 11.
    • When you solve for y, you'll get: y = -2x + 11
  3. 3
    Substitute this expression for y into the second equation. Simply enter those values into your other equation, within parentheses, in the spot where y was. Make sure you add parentheses so that any coefficients are applied to the whole value for y.
    • In this example, you'll get 3x - 2(-2x + 11) = 6
  4. 4
    Solve the equation for x. It might get complicated depending on how complex your equations are, but it's doable!
    • When you multiply the equation out, 3x - 2(-2x + 11) = 6 becomes 3x + 4x - 22 = 6.
    • Then you can add the x values and subtract 22 from both sides. You'll get 7x = 28.
    • Divide both sides by 7 to get x = 4.
  5. 5
    Use your x value in an original equation to solve for y. Plugging your x value back in to one of the original equations let you solve for y.
    • For example, 2x + y = 11 becomes 2(4) + y = 11. This can be simplified to 8 + y = 11. Then you can solve for y, and find that y = 3.
  6. 6
    Double check your values in the original equations. If you found the right solution, the x and y values will make both equations true.
    • In this example, 2x + y = 11 comes out to 2(4) + 3 = 11, which is true, and 3x - 2y = 6 comes out to 3(4) - 2(3) = 6, which is also true, so your results for x and y are accurate. Well done!
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