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Solving Systems of Equations - Substitution

SUBSTITUTION means: replacing a variable with something of equal value.


If we know that a variable is equal to a number or expression, then we can replace that variable with the number or expression:


If  x = 4, then we can replace  x  with  “4”.

So, for example,  9x – 3  becomes 9(4) – 3.


If  y = 2x – 5, then we can replace “y” with “2x – 5”.  (Be sure to use parentheses.)

So, for example,  3x + 7y  becomes  3x + 7(2x – 5).


When we have a system of two linear equations, we can use SUBSTITUTION to solve the system.  Follow this procedure:


1)    Make sure one of the equations has an isolated variable.  (We may need to use algebra to isolate a variable.)

2)    Use substitution to replace the variable in the other equation.

3)    Solve the other equation, which now has only one variable (at most).

(If, while solving, all of the variables disappear, then the system is special.  See the tutorial on “special systems”)

4)    Now that we have a number value for one of the variables, we use substitution to replace that variable in either of the original equations.

5)    Solve the equation to find the number value for the final variable.

6)    Report the final answer. (In general, an ordered pair is preferred.)


EXAMPLES WITH EXPLANATIONS:        (Click images to enlarge.)