Substitution Method for Systems of Equations

The substitution method is one of the fundamental techniques used to solve systems of equations, particularly in introductory algebra and linear algebra.

How the method works

Solve one of the equations for one of its variables.
Substitute this expression into the remaining equations.
Continue the process until all variables have been determined.

The procedure continues until the system is fully resolved, leading to one of the standard outcomes: a unique solution, no solution, or infinitely many solutions.

Note. A system is consistent with a unique solution if it has exactly one solution, inconsistent if it has none, and consistent with infinitely many solutions if it admits a whole family of solutions.

The substitution method applies to systems with two or more equations in two or more unknowns. It is especially effective when at least one equation can be rearranged easily.

A worked example

Consider the system:

$$ \begin{cases} 2x + y = 4 \\ 3x + 2y = 2 \end{cases} $$

Begin by isolating $ y $ in the first equation:

$$ \begin{cases} y = 4 - 2x \\ 3x + 2y = 2 \end{cases} $$

Now substitute $ y = 4 - 2x $ into the second equation:

$$ \begin{cases} y = 4 - 2x \\ 3x + 2 \cdot (4 - 2x) = 2 \end{cases} $$

Expand and simplify:

$$ \begin{cases} y = 4 - 2x \\ 3x + 8 - 4x = 2 \end{cases} $$

Combine like terms:

$$ \begin{cases} y = 4 - 2x \\ -x = 2 - 8 \end{cases} $$

Which becomes:

$$ \begin{cases} y = 4 - 2x \\ -x = -6 \end{cases} $$

Thus:

$$ \begin{cases} y = 4 - 2x \\ x = 6 \end{cases} $$

Now substitute $ x = 6 $ back into the expression for $ y $:

$$ \begin{cases} y = 4 - 2 \cdot (6) \\ x = 6 \end{cases} $$

Simplifying gives:

$$ \begin{cases} y = 4 - 12 \\ x = 6 \end{cases} $$

So:

$$ \begin{cases} y = -8 \\ x = 6 \end{cases} $$

Hence the solution to the system is the ordered pair $ (x, y) = (6, -8) $.

Verification. Substitute $ x = 6 $ and $ y = -8 $ into the original system: $$ \begin{cases} 2x + y = 4 \\ 3x + 2y = 2 \end{cases} $$ to obtain $$ \begin{cases} 2 \cdot 6 + (-8) = 4 \\ 3 \cdot 6 + 2 \cdot (-8) = 2 \end{cases} $$ $$ \begin{cases} 12 - 8 = 4 \\ 18 - 16 = 2 \end{cases} $$ $$ \begin{cases} 4 = 4 \\ 2 = 2 \end{cases} $$ Both equations hold, confirming that the solution is correct.

And so on.