Eliminating Variables With Equality Constraints

Why Eliminate a Variable?

In an optimization problem, the objective may be written using several variables, even though an equality constraint ties those variables together. The variables are then not free to vary independently. Substituting the equality constraint into the objective can describe the same choices with fewer variables.

The important requirement is that the substitution must preserve the original feasible choices. A rewritten formula is useful only if it represents exactly the choices allowed by the original problem—not a larger set, a smaller set, or just one part of the set without saying so.

Feasible Choices and a Reduced Objective

Suppose an objective is \(F(x,y)\), and the variables must satisfy an equality constraint \(g(x,y)=0\), along with any other stated conditions. The feasible choices are the pairs \((x,y)\) that satisfy all of those requirements. The objective assigns a value \(F(x,y)\) to each such pair.

If the equality can be solved as \(y=\phi(x)\), substitution gives the one-variable expression \(F(x,\phi(x))\). This expression is called the reduced objective. It describes the objective values along the constraint, provided we also retain the conditions that specify which values of \(x\) correspond to allowed original choices.

This definition highlights two separate tasks. First, solve the equality correctly. Second, preserve all the restrictions that determine which original choices are allowed. Substitution does not remove conditions such as nonnegativity, positivity, or another constraint.

Solving the Equality Carefully

For a linear equality such as \(ax+by=c\), solving for \(y\) requires \(b\ne 0\). In that case, rearranging gives \(y=(c-ax)/b\). If \(b=0\), this rearrangement is not allowed; the equality instead places a condition directly on \(x\), or may be inconsistent.

For nonlinear equalities, solving may produce more than one expression. For example, \(y^2=q(x)\) gives \(y=\sqrt{q(x)}\) or \(y=-\sqrt{q(x)}\) when \(q(x)\ge 0\). If the original problem also requires \(y\ge 0\), only the nonnegative branch is permitted. If no such condition is given, both branches must be considered. Choosing one branch without justification can discard feasible choices.

There is also a difference between substituting into the objective and simplifying the entire problem. Substitution changes how the objective is written; it does not by itself establish which parameter values are allowed or where an optimum occurs. Any remaining conditions must still be respected.

Worked Example: A Linear Resource Constraint

Why the Substitution Preserves the Choices

The reasoning behind substitution can be stated precisely. Suppose the original equality is equivalent, under its stated conditions, to \(y=\phi(x)\). Let \(S\) be the set of original feasible pairs and let \(T\) be the set of parameter values \(x\) that produce pairs satisfying all the original conditions. Define the correspondence \(P(x)=(x,\phi(x))\).

To verify the correspondence, consider both directions. For any \(x\in T\), the definition of \(T\) ensures that \((x,\phi(x))\) satisfies the equality and every other original condition. Thus \(P(x)\in S\). Conversely, take any \((x,y)\in S\). Since it satisfies the equality, the assumed equivalent expression gives \(y=\phi(x)\). Therefore \((x,y)=P(x)\), so every original feasible pair is represented. If two parameters \(x_1\) and \(x_2\) produce the same pair, their first coordinates must agree, so \(x_1=x_2\). The correspondence is one-to-one.

Finally, the reduced objective is defined by substituting the corresponding pair into the original objective, so \(R(x)=F(x,\phi(x))\) by definition. This proves that the substitution preserves both the feasible choices and their objective values. In particular, comparing objective values after substitution is meaningful only when the allowed parameter values represent the original feasible set as described.

Worked Example: A Constraint with Two Algebraic Branches

Worked Example: A Product Constraint

Common Mistake: Dropping a Condition or a Branch

A frequent error is to treat an algebraic expression obtained by rearranging the equality as though it automatically captures all original choices. For example, from \(x^2+y^2=25\), one might write \(y=\sqrt{25-x^2}\) and claim that this represents every pair on the circle. It does not: the square root is nonnegative, so it represents only pairs with \(y\ge 0\).

A correction must reflect the original conditions. If \(y\ge 0\) is part of the problem, the positive square-root expression is appropriate. If there is no sign restriction, both expressions \(y=\sqrt{25-x^2}\) and \(y=-\sqrt{25-x^2}\) must be accounted for. Similarly, when dividing by a variable, first establish that it cannot be zero, or treat the zero case separately.

Why This Step Matters

Eliminating a variable makes the dependence imposed by an equality explicit. It can simplify later analysis by reducing the number of independent variables in the objective. But simplification is not permission to change the problem: the conditions defining the original choices remain part of the mathematical model.

This is especially important when an equality has multiple branches, when solving requires division, or when other restrictions are present. A reliable substitution records the expression used, the conditions that justify it, and any remaining restrictions. With those details preserved, the reduced objective can be used to study the same choices as the original formulation.