Equations, identities and rearrangement · Extension
הוצג| English | משמעות |
|---|---|
| identity/aɪˈdentɪti/ | זהות |
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.
The method
- 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.
What distinguishes an identity from an equation?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
x+x=2x is true for every x, so it is an identity; 2x=6 is true only for x=3, so it is an equation asking for that solution.
A worked case
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
An equation asks which inputs satisfy an equality; an identity holds for all allowed inputs
Compare the model with the worked case and explain one change.
Solve 5x-7=18.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Add 7 then divide by 5: x=(18+7)/5=5.
Where it goes wrong
- 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.
Not true: Cancelling x from (x+3)/x leaves 3 for every nonzero x. x is a term of the numerator, not a factor, so it cannot be cancelled: at x=1, (1+3)/1=4, not 3.
Solve 2(x+3)=18.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Divide by 2 then subtract 3: x=9-3=6.
Cancelling x from (x+3)/x leaves 3 for every nonzero x.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
x is a term of the numerator, not a factor, so it cannot be cancelled: at x=1, (1+3)/1=4, not 3.
Applying it
- 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.
Find the cost of 20+3x when x=12.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Substitute the usage: 20+3×12=56.
Match each operation to the rule that keeps it valid.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Equality is preserved only by applying the same valid operation to both sides; dividing by zero is never valid.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.2.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Identity: An equality that holds for every allowed value of its variable.
- An equation asks which inputs satisfy an equality; an identity holds for all allowed inputs.
- Watch for: cancelling a term is not the same as cancelling a factor.
Write from the solution x equals 8
- Solve 5(x - 2) = 3x + 6.
- Say how an identity differs from this equation.
Solve 5(x - 2) = 3x + 6. How is an identity such as 2(x + 3) = 2x + 6 different?
5 times 6 is 30, and 3 times 8 plus 6 is 30.
What is x?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
2x = 16, so x = 8. This equation is not an identity. Other options — 12 (the lesson solution of a different linear equation); Every number (because this equation is an identity).