Write a rule for a $10 fee plus $4 per item
- Let n be the item count and C the cost in dollars.
- The repeated charge is 4n; the fixed charge is 10.
- C = 4n + 10. At n = 3, cost = $22.
Writing a rule from evidence
Build an equation from a context, a table or a graph—and test it.
Lesson 08 · Start where you need
Your learning goal
Identify m and b, define variables and verify a linear rule using known pairs.
Choose the input and output
Let h be rental time in hours and C be total cost in dollars. If the charge is $8 once plus $3 for each hour, the rule is C = 3h + 8. The coefficient has units of dollars per hour. The constant has units of dollars. Define these meanings before substituting numbers.
Words worth knowing
- Ordered pair
- An input and its output, written (x, y).
- Rate of change
- Change in output divided by change in input.
- Starting value
- The output when the input is zero.
- Domain
- The set of permitted inputs for the situation.
1
Make the idea visible.
Predict a change, adjust one value, then explain what stayed the same.
The line represents a rule. Use the context to decide which inputs are meaningful.
y = 3x + 8
The same rule in a table
| x | y |
|---|
Use a table to find the rate
From the pairs (2, 11) and (5, 20), the rate is (20 − 11) ÷ (5 − 2) = 3. Check other rows before treating the whole table as linear. One rate calculated from two points does not prove all additional points belong to the same line.
Recover a missing starting value
Use y = 3x + b and one known pair. With (2, 11), write 11 = 3(2) + b, giving b = 5. The rule is y = 3x + 5. Substitute the other pair: 3(5) + 5 = 20. Agreement checks the calculation.
Keep the rule separate from its domain
A school printing order may cost C = 2n + 12 for n whole booklets. The rule describes the price structure, while the domain states which n-values are available. A maximum order size, whole-number restriction or one-time fee can matter when interpreting the graph.
Khan Academy
Watch with a purpose
Linear equations in slope-intercept form
Identify the rate and the starting value before the explanation. Check whether the example’s input and output have units.
Watch on YouTube →Use captions when available. The lesson’s explanations and worked examples cover the idea in text.Worked examples
Follow the reasoning.
Find the rule through (1, 8) and (4, 17)
- Rate m = (17 − 8) ÷ (4 − 1) = 9 ÷ 3 = 3.
- Use (1, 8): 8 = 3(1) + b, so b = 5.
- y = 3x + 5. Check (4, 17): 3(4) + 5 = 17.
Try it with support
Show your thinking.
Write a rule with rate 5 and output 17 at x = 2.
Give me a hint
Substitute the known pair into y = 5x + b.
Compare a worked response
y = 5x + 7
17 = 10 + b, so b = 7.
Explain why y = 4x + 6 matches (3, 18).
Give me a hint
Evaluate the right side at x = 3.
Compare a worked response
4(3) + 6 = 18.
The calculated output equals the pair’s y-coordinate.
Independent practice
Check your understanding.
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Check the concepts, too.
Choose a response, then use the explanation to check your reasoning. This is practice, not a school mark.
Go further
Keep the connections close.
Khan Academy: graphing lines & slope → · All related learning collections →