C2.5; E2.5 - Simultaneous linear equations

A solution that fits both equations

Two numbers add to nine, and their difference is three. What are they? The first clue alone allows $1$ and $8$, $2$ and $7$, and many more pairs. The second clue must be true for the same pair.

Let the numbers be $x$ and $y$. The clues become $x+y=9$ and $x-y=3$. Keeping the letters in the same roles matters: $x-y$ describes how much $x$ exceeds $y$.

You can test a proposed pair by substituting it into each equation separately. Passing one check is necessary, but it is not enough.

Keep the equations but change the proposed pair. This pair still totals nine, so the second equation decides whether it works.

A pair can be displayed as $(x,y)=(6,3)$. The order tells you which number belongs to each unknown. Swapping to $(3,6)$ changes $x-y$ to $-3$, so it does not solve these equations.

Substitution checks a proposed pair without requiring you to solve the equations again.

Eliminating an unknown

Elimination removes one unknown by combining the two equations. If coefficients have equal size and opposite signs, add; if they are equal with the same sign, subtract. Why is adding equations valid? Each left side equals its corresponding right side, so the sums of the two left sides and the two right sides are equal.

After eliminating one unknown, solve the resulting one-unknown equation. Then substitute that value into either original equation to recover the other unknown.

Double the whole second equation, leaving the same solution pair but changing its appearance. The $y$ coefficients no longer cancel straight away. Multiply the whole first equation by two to prepare matching coefficients.

Doubling an equation preserves its solutions. Keeping the same solution pair lets you focus on the extra preparation step. In another problem you may need to multiply both equations, for example by three and two to match coefficients of two and three.