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Teaching video
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Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Use three independent pairwise conditions to recover three unknown quantities.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Three jars hold nonnegative amounts p, q, and r. Their pairwise totals are p+q=9, q+r=7, and p+r=8. Find each amount.
Why this math matters
Combine independent totals to recover individual quantities, then check feasibility as well as algebraic consistency. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recover three quantities from pairwise totals
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Three jars hold nonnegative amounts p, q, and r. Their pairwise totals are p+q=9, q+r=7, and p+r=8. Find each amount.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
2p+2q+2r=9+7+8=24
Adding the three conditions counts each jar twice.
Develop the calculation
p+q+r=12
Divide the combined total by two before recovering individual amounts.
Check and interpret
p=12−7=5; q=12−8=4; r=12−9=3
Each amount is the complete total minus the other pair's total; all three original equations check and the amounts are nonnegative.
The result
p=12−7=5; q=12−8=4; r=12−9=3
Each amount is the complete total minus the other pair's total; all three original equations check and the amounts are nonnegative.
Common mistakes to catch
- The sum of pairwise totals counts every quantity twice.
- A unique real solution can still be impossible under a nonnegative physical-amount restriction.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find a, b, and c if a+b=8, b+c=10, and a+c=12.
Show a hint
Half the sum of the pairwise totals is the complete total.
Reveal answer and explanation
a=5, b=3, c=7
The total is fifteen; subtract ten, twelve, and eight respectively to recover each amount.
Practice 2
Could nonnegative jar amounts have pairwise totals p+q=2, q+r=3, and p+r=9?
Show a hint
Compute the total first, then recover q.
Reveal answer and explanation
No
The total would be seven, giving q=7−9=−2; that real algebraic solution violates the nonnegative-amount condition.
Take the idea with you
Combine independent totals to recover individual quantities, then check feasibility as well as algebraic consistency.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Eliminate a variable with matched coefficients
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