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Method of Substitution (Four-Step Method)

Although on the test you can usually solve a system of two equations in two unknowns by merely adding or subtracting the equations, you still need to know a standard method for solving these types of systems.
 

The four-step method will be illustrated with the following system:

2x + y = 10

5x – 2y = 7

  1. Solve one of the equations for one of the variables:
    Solving the top equation for y yields y = 10 – 2x.
  2. Substitute the result from Step 1 into the other equation:
    Substituting y = 10 – 2x into the bottom equation yields 5x – 2(10 – 2x) = 7.
  3. Solve the resulting equation:
    5x – 2(10 – 2x) = 7
    5x – 20 + 4x = 7
    9x – 20 = 7
    9x = 27
    x = 3
  4. Substitute the result from Step 3 into the equation derived in Step 1:
    Substituting x = 3 into y = 10 – 2x  yields y = 10 – 2(3) = 10 – 6 = 4.
Hence, the solution of the system of equations is the ordered pair (3, 4).
 
Note: Students often mistakenly add 2x to both sides of this equation because of the minus symbol between 2x and 10. But 2x is positive, so we subtract it. This can be seen more clearly by rewriting the right side of the equation as –10 + 2x.




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