PAPER · REASONING · PRACTICE

Linear models: equations, inequalities and intersections

Translate contextual conditions and connect graphical solutions with exact algebra.

40 minutes · Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37

Textbook reference: Year 8, lower volume · Chapter 19: Linear functions · lessons 34–37

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: Predict when plans A=2n and B=n+6 cost the same and which is cheaper for small whole-number usage. 8 min: Use the existing two-plan tariffs inquiry: fixed fee 6, rate 1, minimum usage 0 and budget 100; or draw both given lines on paper. Record n=4,6,8 and costs (8,10),(12,12),(16,14). The one-line inquiry cannot display both plans simultaneously and its intercept control does not reach 6. 5 min: Explain the crossing (6,12), the cost inequality on each side and n as a nonnegative whole number. Connecting lines help comparison; a fractional use is not a valid count. 2 min: Individually choose the cheaper plan at n=7, with both calculated costs and units.

Organise the key idea

A zero solves the corresponding equation. Graphs above or below an axis describe inequalities; a common point satisfies both original equations.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

A delivery costs a fixed 6 dollars plus 2 dollars per item. Write C in terms of item count n and find the cost of 5 items.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The fixed fee is paid once; the item charge is 2n.
C(5)=2×5+6=16.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

For a nonnegative whole-number count n of uses, plan A costs 2n dollars and plan B costs n+6 dollars. Find the equal-cost point and explain its two coordinates.

Equal costs: $2n=\square+6$ .
Subtract n: $n=\square$ ; cost = $2\times\square=\square$ .
Intersection (number of uses, cost) = (____, ____).

Continuous comparison lines over the displayed window 0–8; actual numbers of uses are nonnegative whole numbers.

At the crossing, the two plans have equal costs; do not add their prices.
Set 2n=n+6, then subtract n from both sides.
Use the resulting n=6 in both 2n and n+6; the two costs must agree.

Equal costs give $2n=n+6$ , so $n=6$ .
6 is a nonnegative whole number, so it is an allowed count. Both plans cost 12 dollars.
The crossing means 6 uses and a 12-dollar cost, not 12 uses and 6 dollars.

Check your work

Equate the two original cost expressions and verify the common cost.
Check that the count is a nonnegative whole number and interpret both coordinates with units.

Prerequisite3

Find the x-intercept and explain why y is set to zero.

y=3x-6

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Every point on the x-axis has y=0.
Solve $3x-6=0$ to obtain $x=2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

For the given function, solve $y>0$ . Explain the required part of the graph.

y=2x-4

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The zero is $x=2$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Profit P=3n−12 dollars for a whole-number sales count n≥0. Find all counts with no loss.

y=3x-12

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The zero is $x=4$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Equate the two expressions for y.
Solve for x, then substitute into both lines to verify y.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Find the x-intercept and explain why y is set to zero.

y=-2x+6

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Every point on the x-axis has y=0.
Solve $-2x+6=0$ to obtain $x=3$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A tank starts with 5 L and gains 2 L/min for 10 minutes. Write V(t), its stated time domain, and V(10).

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Include the initial 5 L; time is restricted to the stated 10-minute stage.
Substitute t=10 to obtain 25 L.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

For the given function, solve $y\geq0$ . Explain the required part of the graph.

y=-1x+4

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The zero is $x=4$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the x-intercept and explain why y is set to zero.

y=4x-8

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Every point on the x-axis has y=0.
Solve $4x-8=0$ to obtain $x=2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

For whole-number uses n≥0, plan A costs 2n+5 and B costs 3n. When is A strictly cheaper?

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

A−B=5−n; require 5−n<0.
Thus n>5, and the whole-number restriction starts at 6.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Equate the two expressions for y.
Solve for x, then substitute into both lines to verify y.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

For the given function, solve $y\leq0$ . Explain the required part of the graph.

y=2x-6

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The zero is $x=3$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

For a whole-number input n≥0, y=n−3. Find the smallest input giving y>0.

y=1x-3

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

The zero is $x=3$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Equate the two expressions for y.
Compare slopes and intercepts after simplifying.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Equate the two expressions for y.
Compare slopes and intercepts after simplifying.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Linear functions, equations, inequalities and simultaneous equations.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Linear models: equations, inequalities and intersections

Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A delivery costs a fixed 6 dollars plus 2 dollars per item. Write C in terms of item count n and find the cost of 5 items.

Guided attempt2

For a nonnegative whole-number count n of uses, plan A costs 2n dollars and plan B costs n+6 dollars. Find the equal-cost point and explain its two coordinates.

Equal costs: $2n=\square+6$ .
Subtract n: $n=\square$ ; cost = $2\times\square=\square$ .
Intersection (number of uses, cost) = (____, ____).

Continuous comparison lines over the displayed window 0–8; actual numbers of uses are nonnegative whole numbers.

Prerequisite3

Find the x-intercept and explain why y is set to zero.

y=3x-6

Prerequisite4

For the given function, solve $y>0$ . Explain the required part of the graph.

y=2x-4

Linear models: equations, inequalities and intersections

Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Profit P=3n−12 dollars for a whole-number sales count n≥0. Find all counts with no loss.

y=3x-12

Independent practice6

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Independent practice7

Find the x-intercept and explain why y is set to zero.

y=-2x+6

Independent practice8

A tank starts with 5 L and gains 2 L/min for 10 minutes. Write V(t), its stated time domain, and V(10).

Exit question9

For the given function, solve $y\geq0$ . Explain the required part of the graph.

y=-1x+4

Exit question10

Find the x-intercept and explain why y is set to zero.

y=4x-8

Linear models: equations, inequalities and intersections — Further practice

Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37

Further practice11

For whole-number uses n≥0, plan A costs 2n+5 and B costs 3n. When is A strictly cheaper?

Further practice12

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Further practice13

For the given function, solve $y\leq0$ . Explain the required part of the graph.

y=2x-6

Further practice14

For a whole-number input n≥0, y=n−3. Find the smallest input giving y>0.

y=1x-3

Extension15

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Extension16

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Linear models: equations, inequalities and intersections — Teacher answers

Year 8, lower volume · text pp. 67–74 · textbook lessons 34–37

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Linear functions, equations, inequalities and simultaneous equations.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict when plans A=2n and B=n+6 cost the same and which is cheaper for small whole-number usage.
8′ Use the existing two-plan tariffs inquiry: fixed fee 6, rate 1, minimum usage 0 and budget 100; or draw both given lines on paper. Record n=4,6,8 and costs (8,10),(12,12),(16,14). The one-line inquiry cannot display both plans simultaneously and its intercept control does not reach 6.
5′ Explain the crossing (6,12), the cost inequality on each side and n as a nonnegative whole number. Connecting lines help comparison; a fractional use is not a valid count.
2′ Individually choose the cheaper plan at n=7, with both calculated costs and units.

Worked example1

A delivery costs a fixed 6 dollars plus 2 dollars per item. Write C in terms of item count n and find the cost of 5 items.

Reference answer

$C=2n+6,\quad C(5)=16$ dollars.

C=2n+6,\quad C(5)=16

The fixed fee is paid once; the item charge is 2n.
C(5)=2×5+6=16.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

For a nonnegative whole-number count n of uses, plan A costs 2n dollars and plan B costs n+6 dollars. Find the equal-cost point and explain its two coordinates.

Equal costs: $2n=\square+6$ .
Subtract n: $n=\square$ ; cost = $2\times\square=\square$ .
Intersection (number of uses, cost) = (____, ____).

Continuous comparison lines over the displayed window 0–8; actual numbers of uses are nonnegative whole numbers.

Reference answer

$(6,12)$

Equal costs give $2n=n+6$ , so $n=6$ .
6 is a nonnegative whole number, so it is an allowed count. Both plans cost 12 dollars.
The crossing means 6 uses and a 12-dollar cost, not 12 uses and 6 dollars.

Equate the two original cost expressions and verify the common cost.
Check that the count is a nonnegative whole number and interpret both coordinates with units.

Continuous comparison lines over the displayed window 0–8; actual numbers of uses are nonnegative whole numbers.

Prerequisite3

Find the x-intercept and explain why y is set to zero.

y=3x-6

Reference answer

$x=2$

x=2

Every point on the x-axis has y=0.
Solve $3x-6=0$ to obtain $x=2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

For the given function, solve $y>0$ . Explain the required part of the graph.

y=2x-4

Reference answer

$x>2$

x>2

The zero is $x=2$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Profit P=3n−12 dollars for a whole-number sales count n≥0. Find all counts with no loss.

y=3x-12

Reference answer

$n\in\{4,5,6,\ldots\}$

n\in\{4,5,6,\ldots\}

The zero is $x=4$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Reference answer

$(2,5)$

Equate the two expressions for y.
Solve for x, then substitute into both lines to verify y.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Independent practice7

Find the x-intercept and explain why y is set to zero.

y=-2x+6

Reference answer

$x=3$

x=3

Every point on the x-axis has y=0.
Solve $-2x+6=0$ to obtain $x=3$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A tank starts with 5 L and gains 2 L/min for 10 minutes. Write V(t), its stated time domain, and V(10).

Reference answer

$V=2t+5,\quad0\leq t\leq10,\quad V(10)=25$ L.

V=2t+5,\quad0\leq t\leq10,\quad V(10)=25

Include the initial 5 L; time is restricted to the stated 10-minute stage.
Substitute t=10 to obtain 25 L.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

For the given function, solve $y\geq0$ . Explain the required part of the graph.

y=-1x+4

Reference answer

$x\leq4$

x\leq4

The zero is $x=4$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Find the x-intercept and explain why y is set to zero.

y=4x-8

Reference answer

$x=2$

x=2

Every point on the x-axis has y=0.
Solve $4x-8=0$ to obtain $x=2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

For whole-number uses n≥0, plan A costs 2n+5 and B costs 3n. When is A strictly cheaper?

Reference answer

$n\in\{6,7,8,\ldots\}$

n\in\{6,7,8,\ldots\}

A−B=5−n; require 5−n<0.
Thus n>5, and the whole-number restriction starts at 6.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Reference answer

$(3,7)$

Equate the two expressions for y.
Solve for x, then substitute into both lines to verify y.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Further practice13

For the given function, solve $y\leq0$ . Explain the required part of the graph.

y=2x-6

Reference answer

$x\leq3$

x\leq3

The zero is $x=3$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

For a whole-number input n≥0, y=n−3. Find the smallest input giving y>0.

y=1x-3

Reference answer

n=4

The zero is $x=3$ .
Use the slope sign to select the correct side, and include the zero only for a non-strict inequality.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Reference answer

No common point: the distinct lines are parallel.

Equate the two expressions for y.
Compare slopes and intercepts after simplifying.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Extension16

Find the common point of the two given lines, or explain why there is none or there are infinitely many.

Use the stated conditions and the labelled scales.

Reference answer

Every point on the same line is common.

Equate the two expressions for y.
Compare slopes and intercepts after simplifying.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.