VertexMath · MTH1W · Free worksheet

Linear Relations Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 50 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Slope: $m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}$ , with $y$ on top. A horizontal line has $m=0$ ; a vertical line has an undefined slope.
Linear test: first differences are constant when the $x$ -values go up by the same step. Rate of change $=\dfrac{\Delta y}{\Delta x}$ , not $\Delta y$ alone.
$y=mx+b$ : isolate $y$ first. $m$ multiplies $x$ ; $b$ is the $y$ -intercept (the value of $y$ when $x=0$ ). Direct variation has $b=0$ .
Parallel: equal slopes. Perpendicular: slopes multiply to $-1$ , so flip and change the sign: $4\to-\frac{1}{4}$ .
Inequalities: dashed line for $<$ or $>$ , solid for $\le$ or $\ge$ ; test a point to choose the side. Dividing by a negative flips the sign.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

First Differences · Slope · y = mx + b · Equations of Lines · Parallel & Perpendicular

K1 Linear or Not?

For each table, use first differences to decide whether the relation is linear. If it is linear, state whether it is a direct or a partial variation and write its equation.

a)
Table (a)

$x$ $-2$ $-1$ $0$ $1$ $2$
$y$ $7$ $4$ $1$ $-2$ $-5$
b)
Table (b)

$x$ $0$ $1$ $2$ $3$ $4$
$y$ $1$ $2$ $5$ $10$ $17$
c)
Table (c)

$x$ $0$ $3$ $6$ $9$
$y$ $0$ $6$ $12$ $18$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$7$	$4$	$1$	$-2$	$-5$

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$1$	$2$	$5$	$10$	$17$

$x$	$0$	$3$	$6$	$9$
$y$	$0$	$6$	$12$	$18$

Stuck? Peek at a hint

First check that the $x$ -values go up by the same step, then subtract consecutive $y$ -values.

K2 Find the Slope

a)
Find the slope of the line through $(-3,4)$ and $(5,-2)$ .
b)
Find the slope of the line through $(-2,6)$ and $(4,6)$ , and of the line through $(3,-1)$ and $(3,5)$ .
c)
Find the slope of the line graphed below. Read both axis scales.

Stuck? Peek at a hint

Write $m=\dfrac{y_2-y_1}{x_2-x_1}$ with every negative coordinate in brackets. In (c), read each point's coordinates from the axis numbers.

K3 Slope and y-Intercept

a)
State the slope and $y$ -intercept of $y=7-3x$ .
b)
State the slope and $y$ -intercept of $2y=x-8$ .
c)
Write the equation of the line graphed below in the form $y=mx+b$ . Each grid square is 1 unit.

Stuck? Peek at a hint

Rewrite each equation as $y=\dots$ first; $m$ is the number that multiplies $x$ . On the graph, look where the line crosses the $y$ -axis.

K4 Equation of a Line

a)
Determine the equation of the line with slope $-3$ that passes through $(3,-4)$ .
b)
Determine the equation of the line through $(-2,7)$ and $(4,-2)$ .
c)
Write $3x-4y=12$ in the form $y=mx+b$ . Then state its slope and $y$ -intercept.

Stuck? Peek at a hint

Substitute the point into $y=mx+b$ and solve for $b$ . In (c), subtract $3x$ from both sides, then divide every term by $-4$ .

K5 Parallel & Perpendicular

a)
State the slope of a line parallel to $y=-\dfrac{2}{5}x+1$ , and of a line perpendicular to it.
b)
Determine the equation of the line perpendicular to $y=4x-3$ that passes through $(8,1)$ .
c)
Determine the equations of the lines through $(-3,2)$ that are parallel to and perpendicular to $y=5$ .

Stuck? Peek at a hint

For a perpendicular slope, flip the fraction and change its sign; write $4$ as $\frac{4}{1}$ first. For (c), sketch $y=5$ .

Real World (A)

Rate of Change · Linear Models · Inequalities in Context

A1 Spring Ice

In spring, the ice on a lake in Muskoka thins at a steady rate. The graph shows the ice thickness $T$ , in centimetres, $d$ days after March 15.

a)
Find the rate of change of the ice thickness. Include units, and say what the sign means.
b)
Write an equation for $T$ in terms of $d$ . How many days after March 15 is the ice 15 cm thick?
c)
The equation gives $T=-9$ when $d=30$ . What does this tell you about the ice on that day?

Stuck? Peek at a hint

Pick two points where the line crosses grid intersections and read their coordinates from the axis numbers.

A2 Parcel Pricing

A Winnipeg courier charges by mass. The table shows the cost $C$ , in dollars, to ship a parcel of mass $m$ , in kilograms.

$m\text{ (kg)}$	$2$	$4$	$7$	$10$
$C\text{ (\$)}$	$13$	$17$	$23$	$29$

a)
Show that the relation is linear, and write an equation for $C$ in terms of $m$ .
b)
What do the $2$ and the $9$ in your equation mean? Find the cost to ship a 15 kg parcel.
c)
The courier does not ship parcels over 30 kg. A customer has $75 for one parcel. What is the heaviest parcel they can ship?

Stuck? Peek at a hint

Find the change in cost and the change in mass between each pair of neighbouring columns.

A3 Trip Fundraiser

A Grade 9 class in Thunder Bay sells pizza slices for $3 each and drinks for $2 each. They need to raise at least $600 for a trip. Let $p$ be the number of slices and $d$ the number of drinks sold.

a)
Write an inequality for the class's goal. Is its boundary line drawn solid or dashed?
b)
Do 150 slices and 70 drinks reach the goal? Do 120 slices and 120 drinks?
c)
The point $(220,-30)$ satisfies your inequality. Does it describe a possible result? State the extra conditions $p$ and $d$ must meet.

Stuck? Peek at a hint

Write the money raised from $p$ slices and $d$ drinks, then compare it with $600.

Think It Through (T)

Error Analysis · Building a Rectangle

T1 Spot the Flaw

A student graphs $4x-2y>6$ :
" $-2y>-4x+6$ , so $y>2x-3$ . I draw a solid line through $(0,-3)$ with slope $2$ and shade above it."
The student's boundary line is shown. Each grid square is 1 unit.

a)
Test the point $(0,0)$ in the original inequality. Is it a solution? Is it in the student's shaded region?
b)
Identify the student's two errors. Give the correct inequality in the form $y\ldots$ and describe the correct graph.
c)
Is $(2,1)$ a solution of $4x-2y>6$ ? Explain using the graph.

Stuck? Peek at a hint

Substitute $x=0$ and $y=0$ into the original inequality first.

T2 Build the Rectangle

Points $A(-3,1)$ , $B(1,3)$ and $C(2,1)$ are plotted below. Each grid square is 1 unit.

a)
Show that $\angle ABC$ is a right angle.
b)
Find point $D$ so that $ABCD$ is a rectangle. Then write the equation of line $CD$ .
c)
A student claims: "Two lines whose slopes have opposite signs are perpendicular." Give a counterexample using a line through $B$ .

Stuck? Peek at a hint

Find the slopes of $AB$ and $BC$ first. In a rectangle, opposite sides are parallel.

Boss Level

Optional challenge

Three in a Row

The points $A(1,1)$ , $B(3,k)$ and $C(k,9)$ lie on one straight line.

a)
Find all possible values of $k$ .
b)
For each value of $k$ , write the equation of the line in the form $y=mx+b$ .
c)
The two lines from (b) and the $x$ -axis enclose a triangle. Find its area.

Stuck? Peek at a hint

On one line, the slope from $A$ to $B$ equals the slope from $A$ to $C$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) Linear (first differences all $-3$ ); partial variation; $y=-3x+1$
b) Not linear: first differences $1, 3, 5, 7$ are not constant
c) Linear (first differences all $6$ for an $x$ step of $3$ ); direct variation; $y=2x$

a) $m=-\dfrac{3}{4}$
b) $0$ ; undefined
c) $m=\dfrac{5}{2}$

a) $m=-3$ , $b=7$
b) $m=\dfrac{1}{2}$ , $b=-4$
c) $y=\dfrac{1}{2}x-2$

a) $y=-3x+5$
b) $y=-\dfrac{3}{2}x+4$
c) $y=\dfrac{3}{4}x-3$ ; $m=\dfrac{3}{4}$ , $b=-3$

a) Parallel: $-\dfrac{2}{5}$ ; perpendicular: $\dfrac{5}{2}$
b) $y=-\dfrac{1}{4}x+3$
c) Parallel: $y=2$ ; perpendicular: $x=-3$

Real World

a) $-1.5$ cm per day: the ice gets $1.5$ cm thinner each day
b) $T=36-1.5d$ ; $14$ days (March 29)
c) Nothing real: a thickness can't be negative. The ice is gone at $d=24$ , so the model only applies for $0\le d\le 24$ .

a) The rate is $\frac{4}{2}=\frac{6}{3}=\frac{6}{3}=2$ dollars per kg in every interval; $C=2m+9$
b) $2 per kilogram; $9 fixed fee per parcel; $39
c) $2m+9=75$ gives $m=33$ , which is over the limit, so 30 kg (cost $69)

a) $3p+2d\ge 600$ ; solid
b) No: $\$590<\$600$ . Yes: exactly $600
c) No: the class can't sell $-30$ drinks. $p\ge0$ and $d\ge0$ , and both must be whole numbers.

Think It Through

a) $0>6$ is false, so not a solution; but $0>-3$ is true, so it is in the student's region
b) Dividing by $-2$ must flip the sign, and $>$ needs a dashed line. $y<2x-3$ : dashed line, shade below it.
c) No: $8-2=6$ , and $6>6$ is false. The point lies on the dashed boundary line, which is not part of the solution.

a) $m_{AB}=\dfrac{1}{2}$ , $m_{BC}=-2$ , and $\dfrac{1}{2}\times(-2)=-1$
b) $D(-2,-1)$ ; $y=\dfrac{1}{2}x$
c) e.g. $y=-\dfrac{1}{2}x+\dfrac{7}{2}$ passes through $B$ and has slope $-\dfrac{1}{2}$ ; $\dfrac{1}{2}\times\left(-\dfrac{1}{2}\right)=-\dfrac{1}{4}\ne-1$ , so it is not perpendicular to $AB$ .

Boss Level

Boss

a) $k=5$ or $k=-3$
b) $k=5$ : $y=2x-1$ ; $k=-3$ : $y=-2x+3$
c) $\dfrac{1}{2}$ square unit

Free cheat sheet

Linear Relations Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

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Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedFind the equation of the line through $(-1,7)$ and $(3,-5)$ . (card 5)

1First differences

$\text{rate of change}=\dfrac{\Delta y}{\Delta x}$

Subtract consecutive $y$ -values:

equal steps, constant	linear
equal steps, changing	nonlinear
unequal $x$ -steps	use $\Delta y\div\Delta x$

Example $x$ : $0,2,4,6$ ; $y$ : $5,11,17,23$ : step $2$ , differences $6$ , so $y=3x+5$ .

Trap $x$ goes up by $2$ and $y$ by $6$ : the rate is $6\div2=3$ , not $6$ .

2Direct or partial?

UseThe $y$ -intercept is the starting value.

line through $(0,0)$	direct variation, $y=mx$
starts at $b\ne0$	partial variation, $y=mx+b$

ExampleTaxi: $4 plus $2 per km, so $C=2d+4$ .

Trap $C=2d+4$ : $1$ km costs $6, but $2$ km costs $8, not $12. Only direct variation doubles.

3Slope

$m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}$

rises left to right	$m>0$
falls left to right	$m<0$
horizontal, $y=k$	$m=0$
vertical, $x=h$	undefined

Example $(-2,5)$ and $(4,-1)$ : $m=(-1-5)/(4-(-2))=-6/6=-1$ .

Trap $(1,-3)$ to $(4,6)$ : $m=(6-(-3))/3=3$ , not $(6-3)/3=1$ . Bracket negatives.

4 $y=mx+b$

UseIsolate $y$ first. Graph: plot $(0,b)$ , then rise and run by $m$ . In a table, $b$ is $y$ at $x=0$ .

Trap $y=5-2x$ : $m=-2$ and $b=5$ , not $m=5$ .

5Equation of a line

UseFind $m$ . Put one point into $y=mx+b$ ; solve for $b$ . Check the other point.

Example $(-1,7)$ , $(3,-5)$ : $m=-12/4=-3$ ; $7=-3(-1)+b$ , so $b=4$ : $y=-3x+4$ .

Trap $m=3$ through $(2,-1)$ : $-1=6+b$ , so $b=-7$ , not $2$ or $-1$ .

Trap $2x-5y=10$ : $-5y=-2x+10$ ; divide every term: $y=0.4x-2$ , not $0.4x+10$ .

6Parallel and perpendicular

$\text{parallel: } m_2=m_1$

$\text{perpendicular: } m_2=-\dfrac{1}{m_1}$

UseNegative reciprocals: product $-1$ . Horizontal $\perp$ vertical.

ExamplePerpendicular to $y=3x+1$ through $(6,2)$ : $m=-1/3$ ; $2=-2+b$ , so $y=-x/3+4$ .

TrapPerpendicular to slope $2/3$ : flip and change the sign: $-3/2$ , not $3/2$ or $-2/3$ .

7Linear inequalities

UseTest a point off the line, e.g. $(0,0)$ : true means shade its side.

Isolate $y$ first.

$<$ or $>$	dashed boundary
$\le$ or $\ge$	solid boundary
$y>$ or $y\ge$	shade above
$y<$ or $y\le$	shade below

Trap $-2y<4x-6$ : dividing by $-2$ flips the sign: $y>-2x+3$ , not $y<-2x+3$ .

8Linear models

UseSlope: rate of change, with units. $b$ : starting value. Extrapolate only where the situation still makes sense.

ExampleA hall costs $150 plus $12 per guest: $C=12g+150$ ; $40$ guests cost $630.

TrapTank: $V=300-12t$ gives $-60$ L at $t=30$ . It is empty at $t=25$ , so $0\le t\le25$ .

Equation of a line: first step

Two points Slope formula

Slope and a point Solve for

b

Parallel or

\perp

line Same

m

-1/m

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Core Skills (K)

K1 Linear or Not?

K2 Find the Slope

K3 Slope and y-Intercept

K4 Equation of a Line

K5 Parallel & Perpendicular

Real World (A)

A1 Spring Ice

A2 Parcel Pricing

A3 Trip Fundraiser

Think It Through (T)

T1 Spot the Flaw

T2 Build the Rectangle

Boss Level

Three in a Row

Answer key

Core Skills

Real World

Think It Through

Boss Level

Linear Relations Cheat Sheet

Unit Summary

Build your own

Phone card

1First differences

2Direct or partial?

3Slope

4y=mx+by=mx+by=mx+b

5Equation of a line

6Parallel and perpendicular

7Linear inequalities

8Linear models

Equation of a line: first step

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What is VertexMath?

4 $y=mx+b$