Solving Two Equations by Elimination

The Idea Behind Elimination

In a system of two linear equations, each equation imposes a condition on the same pair of variables. Substitution solves one equation for a variable and uses that expression in the other. Elimination takes a different route: we add or subtract equations, sometimes after scaling them, so the terms involving one variable cancel. The resulting equation has only one variable, which can then be solved directly.

Throughout, let \(x\) and \(y\) be real variables, in that order. A solution to a system is an ordered pair \((x,y)\) that makes both equations true. For example, \(2x+y=7\) and \(x-y=2\) form a system; a value of \(x\) or \(y\) by itself is not a solution to the system.

The cancellation is the key step. If the \(y\)-terms are \(3y\) and \(-3y\), adding the equations removes \(y\). If the terms are \(4x\) and \(4x\), subtracting one equation from the other removes \(x\). When the coefficients do not already match in this way, scale an equation first. Every operation should be applied to a whole equation, including its right-hand side.

Why Combining Equations Is Valid

Elimination is useful only if it keeps the original solution pairs. Multiplying an equation by a nonzero number is reversible: divide by that number to recover the original equation. Adding a multiple of one equation to another is reversible too: subtract the same multiple to undo the addition. These facts explain why elimination can simplify a system without losing its solutions.

Proof. Suppose the equations are \(E_1\) and \(E_2\), and a pair \((x,y)\) satisfies both. If \(E_1\) is multiplied by a nonzero number \(k\), the resulting equality still holds, because both sides of a true equality can be multiplied by \(k\). Conversely, if the scaled equation holds, division by \(k\ne0\) gives \(E_1\). Thus scaling preserves exactly the same solution pairs. Now replace \(E_2\) by \(E_2+cE_1\), where \(c\) is any real number. A pair satisfying the two original equations satisfies this new equation by addition. Conversely, if it satisfies \(E_1\) and \(E_2+cE_1\), subtracting \(cE_1\) from the new equation gives \(E_2\). So it satisfies the original pair of equations as well. In either operation, the solution pairs before and after are the same. \(\square\)

This justification also shows why multiplying an equation by zero is not an allowed scaling step. It would turn any equation into \(0=0\), which cannot in general be reversed to recover the original condition.

A Reliable Elimination Procedure

For instance, to cancel \(y\) from equations with \(y\)-coefficients \(b_1\) and \(b_2\), one can scale the equations to make these coefficients opposites, provided the coefficients are both nonzero. The same idea works for \(x\). If one coefficient is already zero, that variable is absent from that equation, so it may be simpler to eliminate the other variable instead. Always check the coefficients before choosing the operation.

What the Eliminated Equation Tells You

Sometimes combining equations does not produce a nonzero coefficient on the remaining variable. Two important outcomes are \(0=0\) and a false equality such as \(0=1\). They have different meanings, which can be read directly from the definition of a solution.

Proof. The elimination step preserves the solution pairs by the theorem above. If it produces a false equality such as \(0=1\), no values of \(x\) and \(y\) can satisfy that equality, since its two sides are unequal. Therefore no pair satisfies the transformed system, and preservation of solutions means no pair satisfies the original system. If it produces \(0=0\), that equality is true for every pair, so it rules out no pairs and selects none by itself. The other equation remains a condition on any solution; consequently \(0=0\) alone does not determine a solution pair. \(\square\)

Common Mistakes and a Careful Check

Do not cancel terms while forgetting to scale the right-hand side. A common incorrect step in the system \(3x+2y=8\), \(5x-3y=7\) is to multiply the first equation by \(3\) but write \(9x+6y=8\). That is not the result of multiplying the whole equation by \(3\): the right-hand side must also be multiplied. The correct equation is \(9x+6y=24\). Keeping the equality balanced is what makes the step reversible and valid.

Do not stop after finding one variable. In the first worked example, \(x=3\) is not the solution pair; the system asks for values of both \(x\) and \(y\). Substitution into an original equation gives \(y=1\). Then checking both original equations confirms that the pair works.

Check signs in subtraction. In the third example, subtracting \(4x-3y\) from \(4x+10y\) gives \(4x+10y-4x+3y=13y\). The sign of the term \(-3y\) changes because the entire expression is subtracted. Writing the subtraction before simplifying helps prevent a sign error.

A dependable solution therefore has three visible parts: a valid cancellation, a calculation of both variable values, and substitution of the resulting pair into both original equations. Elimination is not complete merely because an equation with one variable has been obtained; the final check is what confirms the pair satisfies the original constraints.