PAPER · REASONING · PRACTICE

Two-variable systems: quantities and models

Name unknowns, keep units consistent, build independent conditions and check contextual feasibility.

40 minutes · Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33

Textbook reference: Year 7 lower · Chapter 8 Simultaneous linear equations · text lessons 31–33

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

18 minutes using tickets: first set prices 2 and 5, total 10, revenue 32; name both unknown counts before revealing equations (3 min). Enumerate and solve, checking nonnegative integer counts and revenue (5 min). Change both ticket prices to the same value 3 while holding total 10: compare revenues 30 and 31 (6 min). Explain “insufficient independent information” versus “contradictory information”, and separate real solutions from feasible ticket counts (4 min). Same-price controls are supported by this existing lab.

Organise the key idea

Name unknowns, keep units consistent, build independent conditions and check contextual feasibility.

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 notebook costs x dollars and a pen y dollars. 2 notebooks and 3 pens cost 16 dollars; 4 notebooks and 1 pens cost 12 dollars. Find both unit prices.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(2)x+(3)y=16\\(4)x+(1)y=12\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(-5)y=-20$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=2$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 32 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Count condition: $x+y=\square$ ; revenue: $2x+5y=\square$ .
Use $x=10-y$ : $2(10-y)+5y=32$ .
Then $3y=32-20=\square$ ; find y=____ and x=____, both whole nonnegative counts.

The total 10 is a count; 32 is revenue in dollars.
Each A ticket contributes 2 dollars and each B contributes 5; use both independent conditions.
Substitute x=10−y into 2x+5y=32, then check count and revenue after solving.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=10\\(2)x+(5)y=32\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(3)y=12$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=6$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

x adults and y children visit a museum. Write their total number. Do not add ticket prices to a count.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both variables count people, so their units match.
The total is x+y.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Convert 1 hour 30 minutes to hours before using a speed in km/h.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

30 minutes is 30/60 hour.
The total is 3/2 hours.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

A tickets cost 3 dollars each and B tickets 5 dollars each. 12 tickets cost 44 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=12\\(3)x+(5)y=44\end{cases}$ .
Equation 2 minus $3$ times equation 1 gives $(0)x+(2)y=8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=8$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

A notebook costs x dollars and a pen y dollars. 3 notebooks and 2 pens cost 19 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(3)x+(2)y=19\\(2)x+(5)y=31\end{cases}$ .
Equation 2 minus $\frac{2}{3}$ times equation 1 gives $(0)x+(\frac{11}{3})y=\frac{55}{3}$ .
Back-substitute to obtain $y=5$ .
Back-substitute to obtain $x=3$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

A boat travels downstream at 28 km/h and upstream at 20 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=28\\(1)x+(-1)y=20\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=24$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

A group has 18 people. Adults number twice the children. Let x and y be adult and child counts. Find both counts.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(-2)y=0\\(1)x+(1)y=18\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(3)y=18$ .
Back-substitute to obtain $y=6$ .
Back-substitute to obtain $x=12$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

If x is a ticket count and p is dollars per ticket, what is px?

The total count in tickets.
A speed.
The total price in dollars.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Multiply dollars per ticket by ticket count.
Ticket units cancel and dollars remain.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Solving a ticket model gives x=11/3, y=7/3. May these be rounded to 4 and 2 and reported as the exact solution? Explain.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

For 6 tickets priced 2 and 5 dollars, the fractional pair gives revenue 19.
The rounded pair (4,2) gives 18 dollars, so it is not a solution.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

A rectangle has perimeter 30 cm. Its length x is 3 cm greater than its width y. Find both dimensions.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=15\\(1)x+(-1)y=3\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-12$ .
Back-substitute to obtain $y=6$ .
Back-substitute to obtain $x=9$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

A notebook costs x dollars and a pen y dollars. 4 notebooks and 3 pens cost 27 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(4)x+(3)y=27\\(2)x+(5)y=31\end{cases}$ .
Equation 2 minus $\frac{1}{2}$ times equation 1 gives $(0)x+(\frac{7}{2})y=\frac{35}{2}$ .
Back-substitute to obtain $y=5$ .
Back-substitute to obtain $x=3$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A boat travels downstream at 34 km/h and upstream at 26 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=34\\(1)x+(-1)y=26\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=30$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Machine A makes x items per hour and B y items per hour. Two hours of A plus one hour of B make 26 items; one hour of A plus three hours of B make 33 items. Find both hourly outputs.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(2)x+(1)y=26\\(1)x+(3)y=33\end{cases}$ .
Equation 2 minus $\frac{1}{2}$ times equation 1 gives $(0)x+(\frac{5}{2})y=20$ .
Back-substitute to obtain $y=8$ .
Back-substitute to obtain $x=9$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Two crops occupy x and y hectares, totalling 100 hectares. Their yields are 2 and 3 tonnes per hectare, with total yield 260 tonnes. Find each crop area, then check both units.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=100\\(2)x+(3)y=260\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(1)y=60$ .
Back-substitute to obtain $y=60$ .
Back-substitute to obtain $x=40$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 60 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Choose one unknown to eliminate from the other equations.
If you multiply an equation, multiply every term and its right side.
Eliminate the same unknown consistently, solve the reduced system, then back-substitute and check all original equations.

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=10\\(2)x+(5)y=60\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(3)y=40$ .
Back-substitute to obtain $y=\frac{40}{3}$ .
Back-substitute to obtain $x=\frac{-10}{3}$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Solve a two-variable system and distinguish counts, unit prices, totals, speeds and consistent time units.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

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 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33

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

Two-variable systems: quantities and models

Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A notebook costs x dollars and a pen y dollars. 2 notebooks and 3 pens cost 16 dollars; 4 notebooks and 1 pens cost 12 dollars. Find both unit prices.

Guided attempt2

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 32 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Count condition: $x+y=\square$ ; revenue: $2x+5y=\square$ .
Use $x=10-y$ : $2(10-y)+5y=32$ .
Then $3y=32-20=\square$ ; find y=____ and x=____, both whole nonnegative counts.

Prerequisite3

x adults and y children visit a museum. Write their total number. Do not add ticket prices to a count.

Prerequisite4

Convert 1 hour 30 minutes to hours before using a speed in km/h.

Two-variable systems: quantities and models

Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

A tickets cost 3 dollars each and B tickets 5 dollars each. 12 tickets cost 44 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Independent practice6

A notebook costs x dollars and a pen y dollars. 3 notebooks and 2 pens cost 19 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Independent practice7

A boat travels downstream at 28 km/h and upstream at 20 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Independent practice8

A group has 18 people. Adults number twice the children. Let x and y be adult and child counts. Find both counts.

Exit question9

If x is a ticket count and p is dollars per ticket, what is px?

The total count in tickets.
A speed.
The total price in dollars.

Exit question10

Solving a ticket model gives x=11/3, y=7/3. May these be rounded to 4 and 2 and reported as the exact solution? Explain.

Two-variable systems: quantities and models — Further practice

Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33

Further practice11

A rectangle has perimeter 30 cm. Its length x is 3 cm greater than its width y. Find both dimensions.

Further practice12

A notebook costs x dollars and a pen y dollars. 4 notebooks and 3 pens cost 27 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Further practice13

A boat travels downstream at 34 km/h and upstream at 26 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Further practice14

Machine A makes x items per hour and B y items per hour. Two hours of A plus one hour of B make 26 items; one hour of A plus three hours of B make 33 items. Find both hourly outputs.

Extension15

Two crops occupy x and y hectares, totalling 100 hectares. Their yields are 2 and 3 tonnes per hectare, with total yield 260 tonnes. Find each crop area, then check both units.

Extension16

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 60 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Two-variable systems: quantities and models — Teacher answers

Year 7, lower volume · text pp. 61–66 · textbook lessons 31–33

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

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: Solve a two-variable system and distinguish counts, unit prices, totals, speeds and consistent time units.

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

Inquiry and discussion: 18 minutes using tickets: first set prices 2 and 5, total 10, revenue 32; name both unknown counts before revealing equations (3 min). Enumerate and solve, checking nonnegative integer counts and revenue (5 min). Change both ticket prices to the same value 3 while holding total 10: compare revenues 30 and 31 (6 min). Explain “insufficient independent information” versus “contradictory information”, and separate real solutions from feasible ticket counts (4 min). Same-price controls are supported by this existing lab.

Worked example1

A notebook costs x dollars and a pen y dollars. 2 notebooks and 3 pens cost 16 dollars; 4 notebooks and 1 pens cost 12 dollars. Find both unit prices.

Reference answer

$x=2,\quad y=4$ .

x=2,\quad y=4

With the stated variable meanings, the two independent conditions give $\begin{cases}(2)x+(3)y=16\\(4)x+(1)y=12\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(-5)y=-20$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=2$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 32 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Count condition: $x+y=\square$ ; revenue: $2x+5y=\square$ .
Use $x=10-y$ : $2(10-y)+5y=32$ .
Then $3y=32-20=\square$ ; find y=____ and x=____, both whole nonnegative counts.

Reference answer

$x=6,\quad y=4$ .

x=6,\quad y=4

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=10\\(2)x+(5)y=32\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(3)y=12$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=6$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

x adults and y children visit a museum. Write their total number. Do not add ticket prices to a count.

Reference answer

$x+y$ people.

x+y

Both variables count people, so their units match.
The total is x+y.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Convert 1 hour 30 minutes to hours before using a speed in km/h.

Reference answer

$\frac{3}{2}$ h.

1+30/60

30 minutes is 30/60 hour.
The total is 3/2 hours.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

A tickets cost 3 dollars each and B tickets 5 dollars each. 12 tickets cost 44 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Reference answer

$x=8,\quad y=4$ .

x=8,\quad y=4

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=12\\(3)x+(5)y=44\end{cases}$ .
Equation 2 minus $3$ times equation 1 gives $(0)x+(2)y=8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=8$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

A notebook costs x dollars and a pen y dollars. 3 notebooks and 2 pens cost 19 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Reference answer

$x=3,\quad y=5$ .

x=3,\quad y=5

With the stated variable meanings, the two independent conditions give $\begin{cases}(3)x+(2)y=19\\(2)x+(5)y=31\end{cases}$ .
Equation 2 minus $\frac{2}{3}$ times equation 1 gives $(0)x+(\frac{11}{3})y=\frac{55}{3}$ .
Back-substitute to obtain $y=5$ .
Back-substitute to obtain $x=3$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

A boat travels downstream at 28 km/h and upstream at 20 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Reference answer

$x=24,\quad y=4$ .

x=24,\quad y=4

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=28\\(1)x+(-1)y=20\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=24$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

A group has 18 people. Adults number twice the children. Let x and y be adult and child counts. Find both counts.

Reference answer

$x=12,\quad y=6$ .

x=12,\quad y=6

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(-2)y=0\\(1)x+(1)y=18\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(3)y=18$ .
Back-substitute to obtain $y=6$ .
Back-substitute to obtain $x=12$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

If x is a ticket count and p is dollars per ticket, what is px?

The total count in tickets.
A speed.
The total price in dollars.

Reference answer

The total price in dollars.

Multiply dollars per ticket by ticket count.
Ticket units cancel and dollars remain.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Solving a ticket model gives x=11/3, y=7/3. May these be rounded to 4 and 2 and reported as the exact solution? Explain.

Reference answer

No. Ticket counts must be integers and rounding can break the original revenue condition.

For 6 tickets priced 2 and 5 dollars, the fractional pair gives revenue 19.
The rounded pair (4,2) gives 18 dollars, so it is not a solution.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

A rectangle has perimeter 30 cm. Its length x is 3 cm greater than its width y. Find both dimensions.

Reference answer

$x=9,\quad y=6$ .

x=9,\quad y=6

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=15\\(1)x+(-1)y=3\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-12$ .
Back-substitute to obtain $y=6$ .
Back-substitute to obtain $x=9$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

A notebook costs x dollars and a pen y dollars. 4 notebooks and 3 pens cost 27 dollars; 2 notebooks and 5 pens cost 31 dollars. Find both unit prices.

Reference answer

$x=3,\quad y=5$ .

x=3,\quad y=5

With the stated variable meanings, the two independent conditions give $\begin{cases}(4)x+(3)y=27\\(2)x+(5)y=31\end{cases}$ .
Equation 2 minus $\frac{1}{2}$ times equation 1 gives $(0)x+(\frac{7}{2})y=\frac{35}{2}$ .
Back-substitute to obtain $y=5$ .
Back-substitute to obtain $x=3$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A boat travels downstream at 34 km/h and upstream at 26 km/h. Let x be its still-water speed and y the current speed, both in km/h. Find x and y.

Reference answer

$x=30,\quad y=4$ .

x=30,\quad y=4

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=34\\(1)x+(-1)y=26\end{cases}$ .
Equation 2 minus $1$ times equation 1 gives $(0)x+(-2)y=-8$ .
Back-substitute to obtain $y=4$ .
Back-substitute to obtain $x=30$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Machine A makes x items per hour and B y items per hour. Two hours of A plus one hour of B make 26 items; one hour of A plus three hours of B make 33 items. Find both hourly outputs.

Reference answer

$x=9,\quad y=8$ .

x=9,\quad y=8

With the stated variable meanings, the two independent conditions give $\begin{cases}(2)x+(1)y=26\\(1)x+(3)y=33\end{cases}$ .
Equation 2 minus $\frac{1}{2}$ times equation 1 gives $(0)x+(\frac{5}{2})y=20$ .
Back-substitute to obtain $y=8$ .
Back-substitute to obtain $x=9$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Two crops occupy x and y hectares, totalling 100 hectares. Their yields are 2 and 3 tonnes per hectare, with total yield 260 tonnes. Find each crop area, then check both units.

Reference answer

$x=40,\quad y=60$ .

x=40,\quad y=60

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=100\\(2)x+(3)y=260\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(1)y=60$ .
Back-substitute to obtain $y=60$ .
Back-substitute to obtain $x=40$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

A tickets cost 2 dollars each and B tickets 5 dollars each. 10 tickets cost 60 dollars. Let x and y be the numbers of A and B tickets. Form and solve a system; check whole nonnegative counts.

Reference answer

No feasible ticket counts. The real solution is $x=-\frac{10}{3},\ y=\frac{40}{3}$ , and 60 dollars exceeds the maximum possible 50.

x=\frac{-10}{3},\quad y=\frac{40}{3}

With the stated variable meanings, the two independent conditions give $\begin{cases}(1)x+(1)y=10\\(2)x+(5)y=60\end{cases}$ .
Equation 2 minus $2$ times equation 1 gives $(0)x+(3)y=40$ .
Back-substitute to obtain $y=\frac{40}{3}$ .
Back-substitute to obtain $x=\frac{-10}{3}$ .
Substitute the complete tuple into every original equation; each left side equals its stated right side.
Interpret both values in the stated units and check any nonnegative or integer restrictions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.