Equations, identities and rearrangement · Équations, identités et réarrangement
| English | Français |
|---|---|
| identity/aɪˈdentɪti/ | identité |
When do two plans cost the same?
- Two mobile plans cost 20+3x and 44+x yuan for x GB. When do they cost the same?
- This lesson studies identity 恒等式: An equality that holds for every allowed value of its variable.
Choose the mathematical structure
- An equation asks which inputs satisfy an equality; an identity holds for all allowed inputs. Preserve equality by applying the same operation to both sides. State restrictions before dividing by a variable.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines identity? · Quelle description définit correctement l'identité ?
An equality that holds for every allowed value of its variable. · Une égalité valide pour toute valeur admissible de sa variable.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
20+3x=44+x gives 2x=24 and x=12. Both plans then cost 56. In A=πr², divide by π and take the positive square root to obtain r=√(A/π), because r is a length.
Equations, identities and rearrangement · Équations, identités et réarrangement
An equation asks which inputs satisfy an equality; an identity holds for all allowed inputs · Une équation demande quels entrées satisfont une égalité ; une identité est vraie pour toutes les entrées autorisées
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
Solve 5x-7=18. · Résoudre 5x-7=18.
Add 7 then divide by 5: x=(18+7)/5=5. · Ajouter 7 puis diviser par 5 : x=(18+7)/5=5.
Test a tempting shortcut
- Cancelling a term is not the same as cancelling a factor. In (x²+2x)/x, factor the numerator and retain x≠0. Check a rearrangement by substitution.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Cancelling x from (x+3)/x leaves 3 for every nonzero x. This claim is false. Explain which definition or assumption it violates.
Solve 2(x+3)=18. · Résoudre 2(x+3)=18.
Divide by 2 then subtract 3: x=9-3=6. · Diviser par 2 puis soustraire 3 : x=9-3=6.
Cancelling x from (x+3)/x leaves 3 for every nonzero x. · Simplifier x dans (x+3)/x donne 3 pour tout x non nul.
Cancelling a term is not the same as cancelling a factor. In (x²+2x)/x, factor the numerator and retain x≠0. Check a rearrangement by substitution. · Simplifier un terme n'est pas la même chose que simplifier un facteur. Dans (x²+2x)/x, mettre en évidence les facteurs du numérateur et conserver x≠0. Vérifier une réarrangement par substitution.
Interpret a new situation
- Set up the equation from units and the meaning of the unknown. A negative or fractional solution may be algebraically correct but impossible for a count.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the cost of 20+3x when x=12. · Trouver le coût de 20+3x lorsque x=12.
Substitute the usage: 20+3×12=56. · Substituer la consommation : 20+3×12=56.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- edexcel IAL pure mathematics; official unit P4. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An equality that holds for every allowed value of its variable. Choose the relationship, show the method, check its assumptions and interpret the result.