topic badge
CanadaON
Grade 9

6.09 Point of intersection

Worksheet
Points of intersection
1

For each of the following, state the coordinates of the point of intersection of the two lines:

a
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
x
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-7
-6
-5
-4
-3
-2
-1
1
2
x
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y
d
-2
-1
1
2
3
4
5
6
7
8
x
-2
-1
1
2
3
4
5
6
7
8
9
y
e
-2
-1
1
2
3
4
5
6
x
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
y
f
-2
-1
1
2
3
4
5
6
7
8
9
x
-2
-1
1
2
3
4
5
6
7
8
9
y
g
-3
-2
-1
1
2
3
4
5
6
7
8
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y
h
-8
-7
-6
-5
-4
-3
-2
-1
1
2
x
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y
i
-4
-3
-2
-1
1
2
3
4
x
-2
-1
1
2
3
4
5
6
7
y
j
-1
1
2
3
4
5
6
7
8
9
10
11
x
-4
-3
-2
-1
1
2
3
4
5
6
y
2

The graph of y = 3 x - 4 is shown on the coordinate plane:

a

Consider the horizontal line with equation y = 8.

State the point of intersection of the graph of y = 3 x - 4 with the line y = 8.

b

Hence determine the value of x that solves the equation 3 x - 4 = 8.

-4
-3
-2
-1
1
2
3
4
x
-2
-1
1
2
3
4
5
6
7
8
9
y
3

The graph of y = -2 x - 4 is shown on the coordinate plane:

a

State the point of intersection of the graph with the line y = - 12.

b

Hence determine the value of x that solves the equation - 2 x - 4 = - 12.

-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-14
-12
-10
-8
-6
-4
-2
2
y
4

The graph of y = -\dfrac{x}{2} + 6 is shown on the coordinate plane:

a

In order to solve the equation \\- \dfrac{x}{2} + 6 = 8, state the equation of the other line that must be graphed on the axes.

b

Hence find the solution to - \dfrac{x}{2} + 6 = 8.

c

Explain why it is not necessary to write the y-value in your answer to part (b).

-6
-5
-4
-3
-2
-1
1
2
3
x
-1
1
2
3
4
5
6
7
8
9
10
y
5

The graph of y = \dfrac{4x}{3} + 5 is shown on the coordinate plane:

a

In order to solve the equation \\ \dfrac{4x}{3} + 5 = 13, state the equation of the other line that must be graphed on the axes.

b

Hence find the solution to \dfrac{4 x}{3} + 5 = 13.

-4
-3
-2
-1
1
2
3
4
5
6
7
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
y
6

Emma wishes to find the point of intersection of the following lines:

y = -2 x + 3 \\ y = -1
a

Complete the table of values for \\y = -2 x + 3:

x-101
y
b

Sketch the graphs of both lines on a coordinate plane.

c

Hence determine the point of intersection.

7

Scott wishes to find the point of intersection of the following lines:

y = 2 x + 1 \\ y = -3 x + 11
a

Complete the table of values for \\y = 2 x + 1:

x-101
y
b

Complete the table of values for \\y = -3 x + 11:

x-101
y
c

Sketch the graphs of both lines on a coordinate plane.

d

Hence determine the point of intersection.

8

For each of the following pairs of equations:

i

Sketch the graph of the two lines on the same coordinate plane.

ii

Find the coordinates of the point of intersection.

a

y = 3

x = - 3

b

y = 2 x + 2

x = - 3

c

y = 3 x - 4

y = 5

d

y = - 2 x + 4

y = 8

e

y = 3 x + 3

x = - 1

f

y = x + 3

y = 3 x - 5

g

y = - x + 6

y = x + 2

h

y = - x + 2

y = 2 x - 4

i

y = - 4 x + 2

y = 3 x - 12

j

y = \dfrac{x}{2} + 3

y = 3 x - 2

k

y = \dfrac{x}{2} - 2

y = - 2 x + 3

l

y = 5 x + 6

y = 2 x + 12

m

y = 3 x - 3

y = - 4 x + 11

n

y = x - 9

y = - x - 7

o

y = 2 x - 7

y = - \dfrac{x}{2} - 2

p

y = \dfrac{x}{2} + 5

y = 3 x

Applications
9

It costs Buzz \$6.00 per month to operate his existing incandescent light bulbs in his home. He has two options moving forward:

  • Option 1 - Incandescent: Stay with the incandescent bulbs at \$6.00 per month.

  • Option 2 - Fluorescent : Buy new energy efficient fluorescent bulbs for \$8.00 which cost \$2.00 per month after they are installed.

The graph shows the cost per month of option 1.

1
2
3
4
5
6
7
8
9
\text{Months}
4
8
12
16
20
24
28
32
36
\text{Cost}
a

On the same graph, plot the relationship for the cost per month of option 2.

b

After how many months would the two options cost the same?

c

After 6 months, which option is cheaper and by how much?

10

A rectangular zone is to be 3 \text{ m} longer than it is wide, with a total perimeter of 18 \text{ m}.

a

Let y represent the length of the rectangle and x represent the width. Construct two equations that represent this information.

b

Sketch the two lines on the same number plane.

c

Hence, find the length and width of the rectangle.

11

A band plans to record a demo at a local studio. The cost of renting studio A is \$250 plus \$50 per hour. The cost of renting studio B is \$50 plus \$100 per hour. The cost, y, in dollars of renting the studios for x hours can be modelled by the linear system:

  • Studio A: y = 50 x + 250

  • Studio B: y = 100 x + 50

a

Sketch the two lines on the same number plane.

b

State the coordinate which satisfies both equations.

c

What does the coordinate from part (d) mean?

12

Michael plans to start taking an aerobics class. Non-members pay \$4 per class. Members pay a \$10 one-time fee, but only have to pay \$2 per class. The monthly cost, y, of taking x classes can be modelled by the linear system:

  • Non-members: y = 4 x

  • Members: y = 2 x + 10

a

Sketch the two lines on the same number plane.

b

State the coordinate which satisfies both equations.

c

What does the coordinate from part (b) mean?

13

The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2 n. The revenue, R, made from selling n toys is given by R = 4 n.

a

Sketch the graphs of cost and revenue on the same number plane.

b

How many toys need to be produced for the revenue to equal the cost?

c

State the meaning of the y-coordinate of the point of intersection.

14

Given the cost function C \left( x \right) = 0.4 x + 2015 and the revenue function R \left( x \right) = 3 x, find the coordinates of the point of intersection, or the break-even point.

155
310
465
620
775
930
1085
1240
x
465
930
1395
1860
2325
2790
y
15

The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.

a

Find the point of intersection of the two equations.

b

Sketch both equations on the same number plane.

c

State the meaning of the point of intersection of the two lines.

16

The two equations y = 4 x + 400 and y = 6 x represent a company's revenue and expenditure respectively.

a

Find the point of intersection of the two equations.

b

Sketch both equations on the same number plane.

c

State the meaning of the point of intersection of the two lines.

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

9.C3.3

Compare two linear relations of the form y = ax + b graphically and algebraically, and interpret the meaning of their point of intersection in terms of a given context.

What is Mathspace

About Mathspace