2.6 - Simultaneous linear equations

Common solutions and elimination

Two numbers have sum $7$ and difference $1$. Can you find both without guessing? One condition leaves many possibilities; using both together will identify the pair.

For example, $(x,y)=(5,2)$ satisfies $x+y=7$, but it does not satisfy $x-y=1$. A pair must pass both checks.

Elimination adds or subtracts complete equations so that one unknown cancels. This works because adding equal quantities to equal quantities gives equal totals. Opposite coefficients cancel by addition; equal coefficients cancel by subtraction. Elimination is not a formula printed on either tier's formulae sheet.

Change only which letter you eliminate. In the same equations, the $x$ coefficients match, so subtraction removes $x$.

Subtraction applies to every term, including any negative terms and both right-hand sides.

Once you find one letter, use an original equation to recover the other.

Scaling equations for elimination

When coefficients do not match, multiply one or both equations to make them match. Multiplying both sides by the same nonzero number preserves every solution, so it is safe to work with the scaled equations.

Choose a common multiple of one letter's coefficients. Multiply every term on each chosen equation, including its right-hand side. Then decide whether addition or subtraction will cancel that letter. No separate scaling formula is supplied on either tier's formulae sheet.

Change the sign pattern so that the scaled coefficients are equal rather than opposite. You must now subtract.

You could eliminate $x$ instead. Choose convenient arithmetic and keep fractions exact; solutions need not be integers.

When Pearson asks for clear algebraic working, show the scaled equations and the elimination; a calculator's final pair does not show that method.

Substitution and paired data

Substitution replaces a letter by an equal expression. It is especially helpful when one equation has already isolated that letter.

If $a+2p=4$, then $a=4-2p$. Replacing $a$ by $4-2p$ preserves its value, so another equation becomes an equation only in $p$. Brackets keep the replacement together. This is an algebraic method, not a result supplied on either tier's formulae sheet.

For a context, define both letters and their units first. Each independent piece of data gives one equation. A mathematically verified surprise is that the quite different purchases below identify each individual price exactly, without buying either fruit on its own.