26 · Audit27 · Build28 · Investigate29 · Present30 · Organise31 · Open
The call Mia has been waiting for…The repaired pump now produces a straight, dependable graph. The dashboard is labelled. Blitz and Luna are settled. Then a rescue group calls about Sol, a gentle horse whose temporary home closes soon.
An empty stall does not automatically mean the stable is ready. Mia must check daily costs, medicine storage, feed limits, fundraising, and paddock space. Each equation is a prediction about Sol's real care. At the end, the numbers must support a clear YES or NOT YET.
An equation such as $y=7x+12$ is a prediction machine. Give it a value of $x$, and it calculates the matching $y$. Work backwards when you know the safe limit or target and need to find the input that reaches it.
The relationship machine
Input
$x=5$
→
Rule
$y=7x+12$
$y=7(5)+12$
→
Output
$y=47$
Keep the variables' jobs visible: $x$ is the chosen input and $y$ is the responding output. Substitute with parentheses so the replacement is easy to see.
Worked example · Solve forwards and backwards
A stable charges $y=7x+12$, where $x$ is hours of pony care and $y$ is total cost in euros.
Forward from $x=5$. $y=7(5)+12=47$, so $5$ hours cost €47.
Backward from $y=68$. Use the balance format:
$$68=7x+12\quad\mid-12$$
$$56=7x\quad\mid\div7$$
$$8=x$$
Interpret. A total cost of €68 corresponds to $8$ hours.
Reasonableness check: $7(8)+12=68$, so the pair $(8,68)$ satisfies the equation.
Practice · Substitute first
Forward: input → output
1 · Sol's temporary-stall cost. Until transport day, the total boarding cost is $y=18x+25$, where $x$ is days and $y$ is euros.
(a) Calculate $y$ when $x=0$.
(b) Calculate $y$ when $x=4$.
(c) Interpret both ordered pairs in context.start with x = 0 → expose the fixed booking cost → substitute a positive input → interpret pairs
2 · Print fundraiser postcards. The total ink used is $y=2.5x+6$, where $x$ is batches printed and $y$ is millilitres.
(a) Determine the ordered pair when $x=8$, showing the substitution that produces it.
(b) State the rate with units.decimal rate → substitute with parentheses → ordered pair → units
3 · Protect Sol's medicine. During a power test, the medicine refrigerator follows $y=24-3x$, where $x$ is hours and $y$ is degrees Celsius.
(a) Calculate $y$ at $x=2$.
(b) Calculate $y$ at $x=7$.
(c) Explain why the outputs decrease.Meaning: the equation predicts the temperature at a chosen time; staff can compare that output with the medicine's safe range.
substitute → preserve negative rate → connect decreasing outputs to −3
Reverse: output → input
4 · Work backwards from the training budget. Sol's settling-in sessions follow $y=8x+17$, where $x$ is visits and $y$ is total cost in euros. Mia can spend €65.
(a) Write the equation with $y=65$.
(b) Solve for $x$ using balance notation.
(c) Interpret the answer as the number of visits the budget buys.replace y, not x → balance operations → check by substitution → interpret
From here, build or compare the relationship yourself.
5 · Feed stock has a real boundary. Use $y=-4x+30$, where $x$ is days and $y$ is bags of feed remaining.
(a) Calculate $y$ for $x=0$, $3$, and $6$.
(b) Solve for $x$ when $y=10$.
(c) Decide whether $(8,-2)$ satisfies the algebraic equation and show the substitution.
(d) Explain why $(8,-2)$ is not a physically sensible feed record even if the substitution is true.forward values → reverse one value → test a claimed pair → separate algebraic truth from physical meaning
6 · When do two fundraising plans match? Plan A follows $y=5x+20$. Plan B follows $y=8x+5$, where $x$ is weeks and $y$ is euros raised.
(a) Write an equation for equal totals.
(b) Solve for the week.
(c) Calculate the equal amount.same y means set rules equal → solve x → substitute back for y
7 · Reach Sol's €390 arrival fund. The stable already has €90 and raises €15 for every sponsored kilometre. The €390 target covers transport, the first veterinary check, and initial feed.
(a) Name $x$ and $y$, including units.
(b) Write the two-variable equation.
(c) Calculate the total after $12$ sponsored kilometres.
(d) Solve how many kilometres produce a total of €390.
(e) Decide READY or NOT YET after $12$ kilometres and calculate the amount still missing, if any.This is the chapter decision: an equation is useful because it tells the team whether a real target has been reached and what must happen next.
story → variables → equation → use it forwards → compare with target → use it backwards
8 · Sensible feed orders. A supplier uses $y=6x+2$, where $x$ is whole pallets and $y$ is feed bags delivered. (a) Calculate $y$ for $x=3$. (b) Explain why $x=3.5$ may not be sensible when the supplier ships only whole pallets, even though the equation can calculate it.equation accepts many numbers → context restricts the valid inputs
Spaced review
9 · Track volunteer preparations. A graph contains $(2,9)$ and $(6,21)$, where $x$ is days and $y$ is arrival tasks completed. Calculate and interpret the rate in tasks per day, showing enough working to verify the coordinate changes you used.signed changes → divide → tasks/day → meaning
10 · Area or perimeter? A $12$ m by $8$ m rectangular paddock needs fencing and grass seed. (a) Calculate the fence length. (b) Calculate the ground area, with correct units.fence traces boundary → seed covers surface → choose units
Show answers
Problem 1a
$y=18(0)+25=25$, so $(0,25)$.
Problem 1b
$y=18(4)+25=97$, so $(4,97)$.
Problem 1c · first pair
At $0$ days the starting cost is €25.
Problem 1c · second pair
Renting for $4$ days costs €97.
Problem 2a
$y=2.5(8)+6=26$, so the ordered pair is $(8,26)$.
Problem 2b
$2.5$ millilitres per batch.
Problem 3a
$y=24-3(2)=18^\circ$C.
Problem 3b
$y=24-3(7)=3^\circ$C.
Problem 3c
The rate is $-3^\circ$C/hour, so temperature falls by $3^\circ$C each hour.
Problem 4b
$$65=8x+17\quad\mid-17$$$$48=8x\quad\mid\div8$$$$6=x$$
Problem 4c
The €65 budget buys $6$ settling-in visits for Sol.
Problem 5a · x = 0
$(0,30)$.
Problem 5a · x = 3
$(3,18)$.
Problem 5a · x = 6
$(6,6)$.
Problem 5b
$$10=-4x+30\quad\mid-30$$$$-20=-4x\quad\mid\div(-4)$$$$5=x$$
Problem 5c
Yes: $-4(8)+30=-32+30=-2$.
Problem 5d
A stable cannot have $-2$ bags. The equation extends beyond the physically meaningful domain, so the stock must be reordered before that point.
Problem 6b
$$5x+20=8x+5\quad\mid-5x$$$$20=3x+5\quad\mid-5$$$$15=3x\quad\mid\div3$$$$5=x$$
Problem 6c
$y=5(5)+20=€45$.
Problem 7a · x
$x=$ sponsored kilometres.
Problem 7a · y
$y=$ total money in Sol's arrival fund, in euros.
Problem 7c
$y=15(12)+90=€270$.
Problem 7d
$$390=15x+90\quad\mid-90$$$$300=15x\quad\mid\div15$$$$20=x$$The stable needs $20$ sponsored kilometres.
Problem 7e
NOT YET. After $12$ kilometres the fund contains €270, so $€390-€270=€120$ is still missing. Reaching $20$ kilometres makes the fund READY.
Problem 8a
$y=6(3)+2=20$ feed bags.
Problem 8b
The context restricts $x$ to whole numbers because the supplier ships only whole pallets.
Problem 9c · calculation
$m=\frac{12}{4}=3$ tasks/day.
Problem 9c · interpretation
The volunteers complete $3$ arrival tasks per day.
Problem 10a
Fence length: $P=2(12+8)=40$ m.
Problem 10b
Ground area: $A=12\times8=96$ m².
Open after checking the answers · The stable story ending
Mia does not confuse “not yet” with “no.” At $12$ sponsored kilometres, the fund is €120 short, so she keeps the event open. At $20$ kilometres, the equation gives exactly €390.
The dimensions fit. The drain and filtered air pass their safety checks. The crew finished before the storm. The repaired pump now has a constant graph. The dashboard records meaningful variables. With the arrival fund complete, Mia can finally call the rescue group back: Sol can come home.
Blitz watches the trailer turn into the new stable. Luna paws at the fresh matting. Mia writes one last ordered pair in the record book—not because the worksheet asks for it, but because the number now tells part of their story.
Story complete · Mathematics as a decision toolAcross Lessons 26–31, the answers audited evidence, shaped a building, diagnosed a fault, justified a grant, organised a dashboard, and made a responsible rescue decision. The next mathematical topic can begin with a new problem in the life of the stable.
