# 3.07 Solving simultaneous equations graphically

Worksheet
Systems of linear equations
1

State the number of solutions for each of the following graphed system of two linear equations:

a
b
c
d
e
2

Consider the following graphs of systems of linear equations:

i

How many solutions does this system of equations have?

ii

Write down the solution(s) to the system of equations.

a
b
c
d
e
f
3

Consider the graph of the equation \\y = 3 x + 2:

If a second line intersects this line at the point \left(0, 2\right), can we determine the equation of the second line? Explain your answer.

4

Consider the graph of the equation \\y = 5 x + 3:

A second line, y = mx + b, intersects this line at the one point (0,3).

a

b

State the value that m cannot be equal to. Explain your answer.

5

A system of linear equations has no solutions. One of the equations of the system is \\y = - 3 x - 2. Determine whether the following could be the other equation of the system.

a

y = - 3 x - 3

b

y = \dfrac{x}{3} - 3

c

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

d

y = 3 x + 2

Solve systems graphically
6

Use the given graph to solve each pair of simultaneous equations:

a
\begin{aligned} y & = x- 2 \\ y & = -8 + 2x \end{aligned}
b
\begin{aligned} y & = x- 2 \\ x-2y & = 4 \end{aligned}
c
\begin{aligned} y & = - 8 + 2 x \\ x - 2y & = 4 \end{aligned}
7

For each pair of linear equations:

i
Sketch the lines on the same number plane.
ii
State the coordinates which satisfies both equations.
a
\begin{aligned} L_1: y &= 4 x - 4 \\ \text L_2: y &= 8 - 2 x \end{aligned}
b
\begin{aligned} L_1: y &= x + 9 \\ L_2: y &= - x - 9 \end{aligned}
c
\begin{aligned} L_1: y &= x + 3 \\ L_2: y &= - x + 3 \end{aligned}
d
\begin{aligned} L_1: 2 x - 4 y &= - 8 \\ L_2: - 4 x + 2 y &= - 8 \end{aligned}
e
\begin{aligned} L_1: y &= 3 x + 6 \\ L_2: y &= - x + 2 \end{aligned}
f
\begin{aligned} L_1: y &= - x - 3 \\ L_2: y &= - 1 \end{aligned}
g
\begin{aligned} L_1: y & = 4x- 3 \\ L_2: y & = 4 - 3x \end{aligned}
8

Consider the following linear equations:

\begin{aligned} L_1: y &= - 2 x - 5 \\ L_2: y &= - 2 x + 4 \end{aligned}
a

Sketch L_1 and L_2 on the same number plane.

b

Is there a coordinate that satisfies the two equations simultaneously? Explain your answer.

Applications
9

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.

10

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?

11

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?

12

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.

13

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.

14

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.

15

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.