39.3 Substitution Method

Substitution Method is an algebraic technique for solving a system of two linear equations by isolating one variable in one equation and replacing that variable everywhere it appears in the other equation, reducing the system to a single equation in one variable that can be solved directly.

Substitution Equation Selection

Procedure

One of the two equations in the system is chosen as the starting point for isolating a variable, typically the equation in which a variable already has a coefficient of one or negative one, since this minimizes the arithmetic needed for isolation.

Example

For the system:

the first equation is selected, since $y$ is already isolated.

One Variable Isolation

Procedure

In the selected equation, one variable is algebraically isolated on one side, expressed entirely in terms of the other variable and constants.

Example

The first equation is already in the required isolated form:

with $y$ fully expressed in terms of $x$.

Isolated Expression Replacement

Procedure

The isolated expression is substituted directly in place of the corresponding variable everywhere it appears in the second, unused equation.

Example

Substituting $2x+1$ in place of $y$ in the second equation:

Substitution Reduced Equation Formation

Procedure

The equation resulting from substitution is simplified by combining like terms, producing a single equation containing only one variable.

Example

Simplifying the previous expression:

Remaining Variable Resolution

Procedure

The reduced one-variable equation is solved using standard algebraic steps, isolating the remaining variable to determine its value.

Example

Solving $5x+1=11$:

Substitution Solution Recovery

Procedure

The value found for the resolved variable is substituted back into the isolated expression from the earlier step, in order to compute the value of the other variable.

Example

Substituting $x=2$ into $y=2x+1$:

Ordered Pair Assembly

Procedure

The two computed values are combined into a single ordered pair, written with the horizontal coordinate first and the vertical coordinate second.

Example

The values $x=2$ and $y=5$ are assembled into the ordered pair $(2,5)$.

Substitution Method Verification

Procedure

Both coordinates of the assembled ordered pair are substituted back into both of the original equations to confirm that each produces a true statement.

Example

Substituting $(2,5)$ into $y=2x+1$ gives $5=5$, and into $3x+y=11$ gives $11=11$, confirming both equations are satisfied and the solution is correct.