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 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to compare two expressions with the same unknown.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Linear equations and substitution.
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
Two fictional rentals cost A(h) = 12 + 3h and B(h) = 6 + 5h dollars for h hours. When do they cost the same, and which is cheaper after that time?
Why this math matters
When a variable appears on both sides, collecting its terms on one side reveals where two linear rules agree. Set two cost or measurement rules equal to find a break-even input, then test either side of it.
Set up the model
A useful answer starts with clear assumptions:
- Both rates stay fixed over the time considered.
- The model permits nonnegative real hours and contains no other charges.
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.
Compare two expressions with the same unknown
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Two fictional rentals cost A(h) = 12 + 3h and B(h) = 6 + 5h dollars for h hours. When do they cost the same, and which is cheaper after that time?
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 quantities
12 + 3h = 6 + 5h
Equal costs mean the two expressions produce the same value for the same input.
Apply the relationship
6 = 2h; h = 3
Subtract 6 and 3h from both sides to isolate the variable.
Check and interpret
A(3) = B(3) = $21; after 3 hours A is cheaper
A has the smaller hourly increase. For example, at 4 hours A is $24 and B is $26.
The result
A(3) = B(3) = $21; after 3 hours A is cheaper
A has the smaller hourly increase. For example, at 4 hours A is $24 and B is $26.
Common mistakes to catch
- Comparing hourly rates alone ignores the starting charges.
- Subtracting a term from only one side breaks equality.
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
Solve 7x + 2 = 4x + 17.
Show a hint
Collect x terms on one side and constants on the other.
Reveal answer and explanation
x = 5
3x = 15 gives x = 5.
Practice 2
For the original rentals, which is cheaper at 1 hour?
Show a hint
Substitute into both full formulas.
Reveal answer and explanation
B is cheaper: $11 versus $15
A(1) = 12 + 3 = 15; B(1) = 6 + 5 = 11.
Take the idea with you
Set two cost or measurement rules equal to find a break-even input, then test either side of it.
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: Recognize equations with no solution or all real solutions
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