Replace equals by equals
Substitution replaces one variable with an equal expression so that only one variable remains.
Learning goals
- Isolate a convenient variable and substitute correctly.
- Back-substitute and verify an ordered pair.
Concept lesson
Make a variable the subject
Choose a variable with coefficient 1 or -1 when possible. From x+y=14, write x=14-y or y=14-x.
Substitution workflow
Isolate, substitute with brackets, solve the one-variable equation, back-substitute, state the ordered pair and verify it in both originals.
Choose the easy isolation
For x+y=14 and x-y=4, isolate x=14-y because no fractions are introduced.
Substitute with brackets
Substitute x=14-y into x-y=4: (14-y)-y=4, hence 14-2y=4 and y=5.
Back-substitute and verify
With y=5, x=14-5=9. Check 9+5=14 and 9-5=4; solution (9,5).
Complete every stage
Solve x+2y=8 and x-y+1=0. The scaffold requires isolation, bracketed substitution, both values and two checks.
Interpret a true or false identity
If substitution ends in 0=0, infer infinitely many solutions; if it ends in 0=5, infer no solution. Link each result to the consistency lesson.
Two-mark substitution structure
Show the isolated expression, substitution equation, first value, back-substituted value and ordered-pair answer.
Exit ticket
Solve 2x+y=10 and x+y=6 by substitution. Justify your isolation choice and verify the ordered pair in both equations.