Substitution and a common solution

A solution of two simultaneous equations is an ordered pair satisfying both. Substitution replaces one unknown with an equal expression from the other equation, leaving one equation in one unknown.

Example

Solve \(y=2x+1\) and \(x+y=10\).

\(\begin{aligned}x+(2x+1)&=10\\3x&=9\\x&=3.\end{aligned}\)

Then \(y=2(3)+1=7\). Check both: \(7=2(3)+1\) and \(3+7=10\). The solution is \((x,y)=(3,7)\).

Substitution is convenient when one variable is already isolated or has coefficient \(1\) or \(-1\). Always give both unknowns.