39.4 Direct Elimination Method

Direct Elimination Method is an algebraic technique for solving a system of two linear equations by aligning the equations so that one variable has coefficients that are exact opposites, then adding the equations together so that variable cancels out entirely, leaving a single equation in one variable.

Linear System Equation Alignment

Procedure

Both equations in the system are rewritten, if necessary, so that like variable terms and the constant term appear in the same relative position in each equation, typically with the $x$ term first, the $y$ term second, and the constant on the opposite side.

Example

For the system:

both equations are already aligned, with matching term order in each row.

Matching Variable Column Placement

Procedure

Once aligned, the equations are visually stacked so that the $x$ terms form one column, the $y$ terms form a second column, and the constants form a third column.

Purpose

This column arrangement makes it visually clear which pairs of coefficients will interact when the equations are combined in the next steps.

Opposite Variable Coefficients

Requirement

For direct elimination to work by simple addition, one variable's coefficients in the two equations must already be exact opposites of one another.

Example

In the aligned system above, the $y$ coefficients are $3$ and $-3$, which are exact opposites, making this variable suitable for direct elimination.

Equation Addition for Elimination

Procedure

When the target variable's coefficients are exact opposites, the two equations are added together term by term, causing that variable to cancel out completely.

Example

Adding the two equations:

which simplifies to $4x=20$, with the $y$ term eliminated.

Equation Subtraction for Elimination

Procedure

When the target variable's coefficients are equal rather than opposite, the two equations are instead subtracted from one another, which also causes that variable to cancel out.

Example

For a system where both equations share the term $5y$, subtracting one equation from the other removes the $y$ term in the same way that addition removes opposite coefficients.

Eliminated Variable Confirmation

Procedure

After adding or subtracting, the resulting equation is checked to confirm that the targeted variable no longer appears, leaving only the other variable and a constant.

Example

The result $4x=20$ contains only $x$, confirming that $y$ was successfully eliminated.

Reduced Elimination Equation Resolution

Procedure

The single-variable equation produced by elimination is solved using standard algebraic steps.

Example

Solving $4x=20$:

Elimination Solution Recovery

Procedure

The value found for the resolved variable is substituted back into either one of the original two equations, and the resulting equation is solved for the remaining variable.

Example

Substituting $x=5$ into $2x+3y=16$:

which solves to $y=2$.

Ordered Pair Completion

Procedure

The two computed values are assembled into a final ordered pair, and both values are substituted into both original equations to confirm the solution is correct.

Example

The values combine into $(5,2)$. Substituting into both original equations gives $16=16$ and $4=4$, confirming the solution satisfies the full system.