Worked example
Example 1: Backsolve before testing choices
Problem
Which choice solves : , , or ?
Undo the fixed first: .
Which choice-testing step preserves the original condition?
Which substitution verifies the selected choice?
Backsolve, Pick Numbers & Eliminate
Use the choices as evidence when direct algebra would waste time.
Ready when you are
7 steps · about 29 minutes
Start with the warm-up.
StartStep 1 of 7Warm-upDone
Warm-up
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Key terms
- Backsolve
- Test answer choices in the original condition.
- Pick numbers
- Replace variables with legal convenient values to test an identity.
- Elimination
- Reject a choice for a specific mathematical reason.
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Try it first
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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.
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Step 3 of 7LearnDone
The big idea
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See the structure
Backsolving starts with the choices, usually the middle one. Substitute it into the original condition and use whether the result is too high or too low to eliminate half the list.
Key idea
Name quantities before calculating
Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.
Build a reliable method
Picking numbers turns an identity with variables into arithmetic. Choose legal values that avoid , , and repeated values, because those can make wrong expressions look right.
Explore in Desmos
Test a billing-model choice
A service charges $4 plus $5 per month for a $19 total. Type the equation, read the vertical solution line at , and substitute 3 to check that .
The calculator is live: change anything and see what happens.
Test a billing-model choice
Degree modePress Escape to leave the calculator (twice if a menu is open).
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Stretch to a 36
Elimination needs a written reason. Cross out choices with the wrong sign, size, unit, or behavior before doing full computation.
Trap
A believable answer to the wrong question
The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.
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Worked examples
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Skip check · 2 questions
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Divide by : .
Worked example
Example 2: Test a constraint without solving everything
Problem
A rectangle has width 5 and perimeter 34. Which proposed length, 10 or 12, satisfies the model?
Use , so .
Which choice-testing step preserves the original condition?
Which substitution verifies the selected choice?
Faded example · you finish it
Example 3: Complete the setup and result
Problem
A plan charges dollars. A $50 bill means how many gigabytes?
Choose the setup that follows the stated relationship.
Choose the resulting value.
Desmos way
Test every proposed value in the original equation
For the choices 7, 9, and 11 in , type
7*7+5, 7*9+5, and 7*11+5 on separate lines.Read 54, 68, and 82. Only the middle output equals the required right side, so . Direct substitution is efficient when the choices are simple.
Test every proposed value in the original equation
Degree modePress Escape to leave the calculator (twice if a menu is open).
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Practice
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Step 6 of 7Spot the mistakeDone
Spot the mistake
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Spot the mistake
Where did it go wrong?
One line has an error. Click the line where it first goes wrong.
What went wrongThis line is correct; the mistake would be stopping without matching the verified value to the requested choice.
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Exit ticket
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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36
These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.