Card 1 of 20
EasyStudy card 1
Prompt
What makes a contextual relationship linear?
Look for a constant additive change.
Answer
Equal changes in the input produce equal changes in the output.
Example
Adding \(12\) dollars each month is linear.
MathChapter 5: Word Problems in Real-Life Situations
Flashcards
Solving Word Problems Using Linear Models Flashcards
Twenty focused cards on model structure, contextual slope and intercept, units, prediction, comparison, and common traps.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
In \(y=mx+b\), what does \(m\) usually represent in context?
It is the coefficient of the input.
Answer
The constant rate of change, measured in output units per input unit.
Example
A slope of \(-4\) liters per minute means losing \(4\) liters each minute.
Card 3 of 20
EasyStudy card 3
Prompt
In \(y=mx+b\), what does \(b\) usually represent?
Set \(x=0\).
Answer
The output when the input is zero: the initial or starting value.
Example
In \(C=9h+25\), \(25\) is the fixed starting charge.
Card 4 of 20
EasyStudy card 4
Prompt
What units should a contextual slope have?
Think rise divided by run.
Answer
Output units per input unit.
Example
Dollars per hour or meters per second.
Card 5 of 20
EasyStudy card 5
Prompt
What does a negative contextual slope mean?
It describes direction, not necessarily a negative output.
Answer
The dependent quantity decreases as the independent quantity increases.
Example
\(-30\) meters per minute indicates descent.
Card 6 of 20
MediumStudy card 6
Prompt
How do you calculate a rate from two observations?
Output change divided by input change.
Answer
Use \(m=(y_2-y_1)/(x_2-x_1)\), keeping subtraction order consistent.
Example
From \((2,50)\) to \((6,90)\), \(m=40/4=10\).
Card 7 of 20
MediumStudy card 7
Prompt
What should you define before writing a word-problem model?
A symbol without a meaning is incomplete.
Answer
Every variable, including what it measures and its units.
Example
Let \(t\) be hours and \(V\) be liters remaining.
Card 8 of 20
MediumStudy card 8
Prompt
How can an unknown initial value be recovered?
Use a point and the rate.
Answer
Substitute a known input-output pair into \(y=mx+b\) and solve for \(b\).
Example
If \(y=5x+b\) and \((3,22)\), then \(b=7\).
Card 9 of 20
MediumStudy card 9
Prompt
How do target-value questions differ from prediction questions?
Do not merely substitute an input.
Answer
Set the output equal to the target and solve for the input.
Example
For \(C=8m+40\) and target \(120\), solve \(120=8m+40\).
Card 10 of 20
MediumStudy card 10
Prompt
How do you evaluate a linear model?
Evaluation starts with a known input.
Answer
Substitute the given input, then simplify while preserving units.
Example
If \(H=3w+10\), then \(H(4)=22\).
Card 11 of 20
MediumStudy card 11
Prompt
What table pattern indicates a linear relationship?
Compare rates between rows.
Answer
Every equal input change has the same output change.
Example
Input increases by \(2\) while output always increases by \(6\).
Card 12 of 20
MediumStudy card 12
Prompt
Why is constant percentage growth generally not linear?
A constant factor is not a constant difference.
Answer
The amount added changes as the current quantity changes.
Example
Growing \(5\%\) yearly usually creates exponential growth.
Card 13 of 20
MediumStudy card 13
Prompt
What is a common rate-versus-change mistake?
Rate is normalized to one input unit.
Answer
Using total change as the per-unit coefficient without dividing by the input interval.
Example
A \(90\)-unit change over \(6\) days is \(15\) units per day.
Card 14 of 20
MediumStudy card 14
Prompt
How should a model from a nonzero observation be used?
Count intervals forward or backward.
Answer
Treat that observation as an anchor; do not mistake it for the intercept.
Example
From day \(3\) to day \(8\) there are \(5\) rate intervals.
Card 15 of 20
MediumStudy card 15
Prompt
How do you find when two linear models match?
A crossover has equal outputs.
Answer
Set their output expressions equal and solve for the shared input.
Example
Solve \(20+5x=50+2x\).
Card 16 of 20
MediumStudy card 16
Prompt
What must be checked before mixing time units in a model?
Hours cannot be multiplied directly by a per-minute rate.
Answer
Convert the input to the unit used by the rate.
Example
\(0.5\) hour is \(30\) minutes.
Card 17 of 20
MediumStudy card 17
Prompt
Why should a linear model's domain be checked?
An algebraic line extends beyond realistic use.
Answer
The context may permit only nonnegative, whole-number, or limited inputs.
Example
A negative number of subscription months is not meaningful.
Card 18 of 20
MediumStudy card 18
Prompt
What does it mean to answer in context?
The variable may be only an intermediate result.
Answer
Translate the computed value back to the requested quantity with units.
Example
Ten may mean ten weeks, not ten dollars.
Card 19 of 20
MediumStudy card 19
Prompt
What is a quick way to check a proposed model?
Verify intercept and rate separately.
Answer
Test the starting input and one later observation.
Example
A model should reproduce every supplied table row.
Card 20 of 20
HardStudy card 20
Prompt
What is the final reasonableness check for a model answer?
Return from algebra to the story.
Answer
Confirm its sign, size, units, domain, and whether it answers the exact question.
Example
A remaining volume cannot exceed the tank's starting capacity in a draining model.
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