39.5 Scaled Elimination Method

Scaled Elimination Method is an extension of the direct elimination technique used when neither variable in a system of two linear equations already has opposite or matching coefficients, requiring one or both equations to be multiplied by a chosen constant so that a variable's coefficients align for elimination.

Elimination Variable Selection

Procedure

One of the two variables in the system is chosen as the target for elimination, typically whichever variable requires the smallest or simplest multipliers to align its coefficients.

Example

For the system:

Coefficient Multiple Identification

Procedure

The least common multiple of the selected variable's two coefficients is identified, since this value determines what each equation must be scaled to before elimination can proceed.

Example

For coefficients $2$ and $4$, the least common multiple is $4$.

Single Equation Scaling

Procedure

When only one equation's coefficient needs adjustment to reach the identified common multiple, that single equation is multiplied through by the required constant.

Example

The first equation is multiplied by $2$ to bring its $y$ coefficient from $2$ to $4$:

producing $6x+4y=32$.

Both Equation Scaling

Procedure

When neither equation's coefficient equals the required common multiple on its own, both equations are multiplied by separate constants chosen so that the selected variable's coefficients become equal in magnitude in both equations.

Example

For a system with $x$ coefficients of $3$ and $5$, the first equation could be multiplied by $5$ and the second by $3$, bringing both $x$ coefficients to $15$.

Equation Equivalence during Scaling

Principle

Multiplying every term of an equation by the same nonzero constant produces a new equation that is mathematically equivalent to the original, meaning it shares exactly the same solution set even though its coefficients have changed.

Verification

Because the scaled equation remains equivalent to the original, any solution found using the scaled version will also satisfy the original, unscaled equation.

Opposite Coefficient Formation

Procedure

If the target variable's scaled coefficients are equal in magnitude but the same sign, one of the two scaled equations is multiplied by negative one, converting the coefficients into exact opposites suitable for elimination by addition.

Example

Continuing the earlier example, the second equation is multiplied by negative one so that $4y$ becomes $-4y$, forming an opposite pair with the first equation's $4y$ term.

Scaled Equation Combination

Procedure

The two scaled equations, now containing opposite coefficients for the target variable, are added together, eliminating that variable and leaving a single equation in the remaining variable.

Example

Adding the scaled equations from the earlier example:

which simplifies to $x=2$.

Reduced Equation Resolution

Procedure

The resulting single-variable equation is solved using standard algebraic steps, matching the procedure used in direct elimination.

Example

The equation $x=2$ is already fully solved, requiring no further steps in this case.

Scaled Elimination Solution Recovery

Procedure

The resolved value is substituted into one of the original, unscaled equations to solve for the remaining variable, and the completed ordered pair is checked in both original equations.

Example

Substituting $x=2$ into $3x+2y=16$ gives $6+2y=16$, which solves to $y=5$, producing the ordered pair $(2,5)$.