C2.5; E2.5 - Simultaneous linear equations and modelling

Elimination

One equation with two unknowns often allows many pairs. For $x+y=7$, the pairs $(1,6)$ and $(2,5)$ both work. A second condition can identify a pair that satisfies both equations at the same time.

Elimination removes one unknown by adding or subtracting equations. You can add equal expressions on the left and equal expressions on the right because equality is preserved. Equal coefficients with the same sign cancel by subtraction; equal coefficients with opposite signs cancel by addition.

When coefficients do not match, multiply an entire equation by a suitable non-zero number. Multiplying just one term would change its solutions. Sometimes one equation needs scaling; sometimes both do. Use the least common multiple of the coefficients you want to match.

If one coefficient already divides the other, scale only the equation that needs it. For example, coefficients $2y$ and $6y$ need the first equation multiplied by 3. Either unknown can be eliminated; choose the smaller useful multipliers.

Display the scaled equations so the cancellation is visible.

Substitution and checking

Substitution replaces an unknown by an expression known to be equal to it. If $y=2x-1$, every occurrence of $y$ can be replaced by $2x-1$. Put that expression into the other equation to create an equation in $x$ alone.

Substituting into the equation just rearranged tells you no new information: it merely repeats an equality. The second equation supplies the extra condition. This method is often convenient when an unknown is already isolated or has coefficient 1.

Now neither equation has an isolated unknown. A coefficient of 1 still makes one rearrangement particularly short.

A substitution check must use the original equations, especially after a bracket expansion or a sign change. Elimination and substitution should agree because they preserve the same two equalities.

Choose the equation with the easier isolated expression.

Building two-equation models

Two unknown quantities need clear meanings before you write two equations. Keep each meaning fixed across both conditions: if $n$ means the price of one notebook in dollars, $2n$ is the price of two notebooks, not their number.

Use one independent condition for each equation. Doubling an equation does not create a new condition; it describes the same relationship again. Tables can keep quantities and totals aligned.