Equations, identities and rearrangement · Higher
도입됨| English | 의미 |
|---|---|
| identity/aɪˈdentɪti/ | 항등식 |
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.
What distinguishes an identity from an equation?
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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.
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
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.
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Add 7 then divide by 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.
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Divide by 2 then subtract 3: x=9-3=6.
Cancelling x from (x+3)/x leaves 3 for every nonzero x.
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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.
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.
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Substitute the usage: 20+3×12=56.
Match each operation to the rule that keeps it valid.
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Equality is preserved only by applying the same valid operation to both sides; dividing by zero is never valid.
Use this in your course
- 4MA1 · Higher · 2. Match the target tier and specification before assigning extensions.
- 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.