MathJewels math guide · Solve linear equations in one variable
Equation Balance and Elimination
Treat equations as balanced statements, transform complete sides together, and eliminate a variable without losing the original system.
The big idea
An equation says that two complete expressions have the same value. If you add, subtract, multiply, or divide only one side, the statement may stop being true. A legal reversible transformation applies the same operation to both complete sides; multiplication or division uses a nonzero number. A system contains two equations that must be true for the same variable values. In elimination, we scale complete equations and add them so one variable has coefficient zero. Keep one original equation beside the new combined equation so the pair still contains enough information to recover both variables.
Worked example
Solve
\[ \begin{aligned} x+y&=7\\ x-y&=1. \end{aligned} \]
Add the complete left sides and complete right sides:
\[ (x+y)+(x-y)=7+1. \]
The (y)-terms cancel, so (2x=8) and (x=4). Keep the first original equation and substitute (x=4): (4+y=7), so (y=3). Check the ordered pair ((4,3)) in both originals: (4+3=7) and (4-3=1).
How children may show it
A learner may use a balance-scale sketch, color-match complete equation sides, write multipliers beside equation labels, align two transformed equations in a vertical addition stack, cross out opposite terms, or substitute the final pair into both originals. The representation succeeds when every factor reaches every term, the combined equation is formed left-with-left and right-with-right, and both checks are visible.
Common mix-up
A common error is multiplying only the variable terms and forgetting the constant, or changing signs on only part of an equation. Ask, “What is the complete left side? What is the complete right side? Where did your factor act on each one?” Another mix-up is keeping only the combined equation. One equation in two variables usually has many solutions, so ask which original equation is still available to recover the other variable.
Try it together
Write two true numerical equalities, such as (8=8) and (5=5). Add their left sides and right sides to make (13=13). Then write a small system whose terms will cancel when added. Before calculating, circle the two terms expected to become zero. After solving, check the ordered pair in both original equations and explain which retained equation supplied the second variable.