# 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, 9* x* – 3 becomes 9(

**4**) – 3.

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

So, for example, 3*x* + 7* y* becomes 3

*x*+ 7(

**2**).

*x*– 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.)

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