Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Connect intersecting, parallel, and coincident lines to algebraic outcomes.
- 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
Classify the system 2x−y=4 and 4x−2y=8.
Why this math matters
Classify the relationship between models before investing in a long solving procedure. 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.
Read a system's solution count from line geometry
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Classify the system 2x−y=4 and 4x−2y=8.
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
Multiply the first equation by 2
It becomes exactly the second equation.
Develop the calculation
Both describe y=2x−4
The two constraints trace the same line.
Check and interpret
Infinitely many real solutions on that line
Every point satisfying one equation satisfies the other.
The result
Infinitely many real solutions on that line
Every point satisfying one equation satisfies the other.
Common mistakes to catch
- Two equations do not guarantee one unique solution.
- Coincident lines are not distinct parallel lines with no intersection.
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
Replace the second right side with 10. How many solutions?
Show a hint
The left side still doubles but the right side does not.
Reveal answer and explanation
None
The equations demand both 4x−2y=8 and 4x−2y=10, which is impossible.
Practice 2
What if the two line slopes are different?
Show a hint
Distinct nonparallel lines in a plane meet once.
Reveal answer and explanation
Exactly one solution
Their unique intersection satisfies both equations.
Take the idea with you
Classify the relationship between models before investing in a long solving procedure.
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: Test whether a table has constant rate
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