read exact points → calculate rate → locate x = 0 → write and verify the equation
Lesson 35.1 · Graph → table → equation
Capture a Line in One Rule
Use exact points to discover the rate and starting value.
A straight graph can describe a horse's feed supply, a travel cost, or the length of a drawing project. A table lets you hold a few exact points still. Then $y=mx+b$ captures every point on the line: $m$ is the signed rate and $b$ is the value when $x=0$.
One relationship, three views
| $x$ | $y$ |
|---|---|
| $0$ | $2$ |
| $1$ | $5$ |
| $2$ | $8$ |
| $3$ | $11$ |
$y=\underbrace{3}_{m}x+\underbrace{2}_{b}$
The table adds $3$ km whenever time adds $1$ hour, so $m=3$. The graph crosses the vertical axis at $(0,2)$, so $b=2$.
Reliable route: read at least two exact points → make a table → calculate $m=\frac{\Delta y}{\Delta x}$ → find $b$ at $x=0$ → write and check $y=mx+b$.
Worked example · Hay remaining
A straight graph contains $(0,18)$, $(2,14)$, and $(5,8)$, where $x$ is days and $y$ is hay in bales.
Table the points. $(0,18)$, $(2,14)$, $(5,8)$.
Find the signed rate. $m=\frac{14-18}{2-0}=\frac{-4}{2}=-2$ bales/day.
Find the start. The point $(0,18)$ gives $b=18$ bales.
Write and check. $y=-2x+18$. At $x=5$, $y=-2(5)+18=8$, matching the graph.
Practice · Build each equation from evidence
Guided: table → m → b → equation
(b) Interpret the negative coefficient in one sentence.
uneven x-steps → verify one shared rate → intercept → equation → units and meaning
two visible points → determine m and b → equation → independent check
From here, decide how to uncover both $m$ and $b$.
(b) Predict $y$ when $x=12$ cm and verify that the prediction follows the graph's pattern.
exact graph evidence → fractional coefficient → equation → extend and verify
(b) Explain why a positive slope and a negative intercept can both be correct.
signed coordinates → positive rise → below-zero starting value → separate m from b
(b) Write Coral's equation from $(0,1)$ and $(3,10)$.
(c) State the coordinate where the plans have equal cost.
(d) Choose the cheaper plan for exactly $2$ rides and calculate the saving.
one equation per line → shared point → compare at a different input → make a decision
Targeted review
known slope = change in y / change in x → solve the denominator equation
balance subtraction → coefficient 3/4 → multiply every side by 4/3
different x-gaps → calculate two quotients → compare
Retention check
percent to decimal → simple interest → principal plus interest
Show answers
Problem 1a
Table points: $(0,4)$, $(1,7)$, $(2,10)$, $(3,13)$.
Problem 1b
$m=\frac{7-4}{1-0}=3$ pages/day.
Problem 1c
$b=4$ pages.
Problem 1d
$y=3x+4$.
Problem 2a
Table points: $(0,25)$, $(2,21)$, $(5,15)$.
Problem 2b
$m=\frac{21-25}{2-0}=\frac{-4}{2}=-2$ kg/day.
Problem 2c
$b=25$ kg.
Problem 2d
$y=-2x+25$.
Problem 2e
The carrot supply decreases by $2$ kilograms each day.
Problem 3a
$m=\frac{23-11}{5-2}=\frac{12}{3}=4$ €/km.
Problem 3b
$$11=4(2)+b$$$$11=8+b\quad\mid-8$$$$3=b$$
Problem 3c
$y=4x+3$.
Problem 3d
$4(5)+3=23$, so $(5,23)$ satisfies the equation.
Problem 4a
Table points: $(0,2)$, $(4,5)$, $(8,8)$.
Problem 4b
$m=\frac{5-2}{4-0}=\frac{3}{4}$.
Problem 4c
$y=\frac{3}{4}x+2$.
Problem 4d
$y=\frac{3}{4}(12)+2=9+2=11$ cm.
Problem 5a
$m=\frac{1-(-3)}{2-0}=\frac{4}{2}=2^\circ$C/hour.
Problem 5b
$b=-3^\circ$C.
Problem 5c
$y=2x-3$.
Problem 5d
The temperature starts below zero at $-3^\circ$C, while the positive slope means it then rises by $2^\circ$C each hour.
Problem 6a
Cedar: $m=\frac{10-4}{3-0}=2$, so $y=2x+4$.
Problem 6b
Coral: $m=\frac{10-1}{3-0}=3$, so $y=3x+1$.
Problem 6c
$(3,10)$, meaning $3$ rides cost €10 on either plan.
Problem 6d · choice
At $x=2$, Cedar costs $2(2)+4=€8$ and Coral costs $3(2)+1=€7$. Choose Coral.
Problem 6d · saving
The saving is $8-7=€1$.
Problem 7a
$\frac{3}{2}=\frac{13-7}{x-2}$.
Problem 7b
$$\frac{3}{2}(x-2)=6\quad\mid\times\frac{2}{3}$$$$x-2=4\quad\mid+2$$$$x=6$$
Problem 8a
$\frac{3}{4}x=9$.
Problem 8b
$$\frac{3}{4}x=9\quad\mid\times\frac{4}{3}$$$$x=12$$
Problem 9a
$\frac{21-9}{4-1}=\frac{12}{3}=4$ bales/delivery.
Problem 9b
$\frac{41-21}{9-4}=\frac{20}{5}=4$ bales/delivery.
Problem 9c
Yes. Both interval rates equal $4$ bales/delivery.
Problem 10a
$I=Prt=600(0.025)(4)=\$60$.
Problem 10b
$\$600+\$60=\$660$.
Next · Lines become tools for predictionYou can now recover $m$ and $b$ from a graph and write its equation. Next, those equations can be compared, extended, and used to find where two real plans meet.