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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Set two expressions equal.
- Collect variable terms without changing equality.
- Interpret the crossing point in context.
Before you start
Distribute brackets, combine like terms, and solve two-step linear equations.
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
Fictional Plan A charges $18 plus $4 per visit. Plan B charges $6 plus $6 per visit. At how many visits do their total costs match?
Why this math matters
Comparing plans means comparing complete totals, not isolated fees. An equation identifies the point where two cost rules meet. Checking counts on either side explains when the lower fixed fee or the lower per-visit rate matters more.
Set up the model
A useful answer starts with clear assumptions:
- Both plans cover the same period and charge their fixed fee once.
- Visits are whole nonnegative counts, and no additional charges apply.
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.
Find when two membership plans cost the same
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Fictional Plan A charges $18 plus $4 per visit. Plan B charges $6 plus $6 per visit. At how many visits do their total costs match?
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.
Compare complete costs
18 + 4v = 6 + 6v
The same variable represents the same visit count for both plans. Equality describes matching total costs.
Collect and solve
18 = 6 + 2v; 12 = 2v; v = 6
Subtract 4v from both sides, subtract six, and divide by two. Each operation preserves the balance.
Check the meeting point
A(6) = 18 + 24 = $42; B(6) = 6 + 36 = $42
Both totals agree. At five visits, A costs $38 and B $36; at seven, A costs $46 and B $48.
The result
The plans match at 6 visits, when either costs $42.
For fewer than six visits, B is cheaper. For more than six, A is cheaper. The $12 difference in fixed fees is offset by A's $2 saving on each visit.
Common mistakes to catch
- Comparing only the per-visit rates ignores the initial fees.
- Moving a term is shorthand for an operation on both sides, not a rule that changes signs without reason.
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
When do 10 + 3n and 4 + 5n dollars match?
Show a hint
Subtract 3n and then subtract four.
Reveal answer and explanation
At n = 3, for $19
10 + 3n = 4 + 5n gives 6 = 2n. Both expressions equal nineteen when n is three.
Practice 2
Solve 4(x + 2) = 2x + 18.
Show a hint
Distribute four before collecting the x terms.
Reveal answer and explanation
x = 5
4x + 8 = 2x + 18 gives 2x = 10. Substitution checks 4(7) = 10 + 18 = 28.
Take the idea with you
When comparing alternatives, calculate the difference between their totals. Its sign tells you which is larger.
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.
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