Topics
2. Linear Relations and Equations
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AustraliaVIC
VCE 11 General 2023

2.05 Simultaneous equations

Lesson
Worksheet
Practice
Worksheet
Graphical method
1

Describe how the graphical solution of a system of two linear equations is found.

2

The following graph displays a system of two equations:

How many solutions does this system of equations have? Explain your answer.

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
3

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

a
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
b
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
c
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
4

The following graph displays a system of two equations:

a

How many solutions does this system of equations have?

b

Write down the solution to the system of equations as an ordered pair \left(x, y\right).

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
5

A system of linear equations has no solutions. If one of the equations is y = - 4 x - 3, choose the other equation in the system out of the following:

A
y = - 4 x - 4
B
y = - \dfrac{x}{4} - 3
C
y = 4 x + 3
D
y = \dfrac{x}{4} - 4
6

For each of the following graphs, write down the solution to the system of equations in the form \left(x, y\right):

a
-3
-2
-1
1
2
3
4
5
x
-7
-6
-5
-4
-3
-2
-1
1
y
b
-1
1
2
3
4
5
6
7
x
-4
-3
-2
-1
1
2
3
4
y
7

The equation y = 5 x + 3 has been drawn on the following graph:

A second line of the form y = mx + b intersects this line at the point \left(0, 3\right).

a

Can the value of b be determined? If so, state the value.

b

Can the value of m be determined? If so, state the value.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
8

Consider the following system of linear equations:

a

Graph the lines on the same number plane.

b

Hence, find the point of intersection of the lines.

\begin{aligned} y & = 4x- 3 \\ y & = 4 - 3x \end{aligned}
9

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

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
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}
Substitution method
10

Solve the following systems of equations using the substitution method:

a
\begin{aligned} y & = 3 x - 18 \\ x +y & = -2 \end{aligned}
b
\begin{aligned} y & = - 6 x - 26 \\ x + y &= - 6 \end{aligned}
c
\begin{aligned} y &= 3 x - 18 \\ x - y &= 10 \end{aligned}
d
\begin{aligned} y &= - 5 x + 22 \\ 6 x + y &= 26 \end{aligned}
e
\begin{aligned} y &= 6 x - 2 \\ - 3 x - y &= - 7 \end{aligned}
f
\begin{aligned} y &= 8 x - 30 \\ 5 x + 9 y &= 38 \end{aligned}
g
\begin{aligned} y &= - 2 x -1 \\ x + 2 y &= 13 \end{aligned}
h
\begin{aligned} 2 x - 5 y &= - 10 \\ y &= - 5 x + 6 \end{aligned}
i
\begin{aligned} y &= 5 x + 34 \\ y &= 3 x + 18 \end{aligned}
j
\begin{aligned} y &= - 5 x - 23 \\ y &= 7 x + 25 \end{aligned}
k
\begin{aligned} y &= 5 x + 12 \\ 4 y &= 28 x + 72 \end{aligned}
l
\begin{aligned} y &= - 4 x - 17 \\ 3 y &= 21 x + 147 \end{aligned}
m
\begin{aligned} y &= 5 x - 8 \\ 2 x - 3 y &= - 15 \end{aligned}
11

Describe how to check whether a given ordered pair is a solution of a system of equations.

12

Determine whether \left(5, 2\right) is a solution of the following system of equations:

\begin{aligned} x + y & = 7 \\ x - y & = 3 \end{aligned}
13

Determine whether \left(4, 17\right) is a solution of the following system of equations:

\begin{aligned} y & = 6x - 7 \\ 4x + 3y & = 67 \end{aligned}
Elimination method
14

For each of the following systems of equations:

i

Describe the steps required to eliminate y.

ii

Hence, solve the system of equations.

a
\begin{aligned} 3x + y &= 4 \\ 2 x - y &= 11 \end{aligned}
b
\begin{aligned} 11 x + 7 y &= -6 \\ 2 x - y &= -17 \end{aligned}
c
\begin{aligned} - 8 x - y &= 5 \\ - 6 x + 3 y &= 5 \end{aligned}
15

For each of the following systems of equations:

i

Describe the steps required to eliminate x.

ii

Hence, solve the system of equations.

a
\begin{aligned} 2 x + 3 y &= 4 \\ x - 2y &= 9 \end{aligned}
b
\begin{aligned} 4 x - 9 y &= 3 \\ -5 x + 7 y &= -8 \end{aligned}
16

For each of the following systems of equations:

i

Describe the steps required to change the fractional coefficients to integer coefficients.

ii

Hence, solve the system of equations.

a
\begin{aligned} \dfrac{4 x}{5} + \dfrac{3 y}{5} &= 7 \\ 8 x - 3 y &= 1 \end{aligned}
b
\begin{aligned} \dfrac{x}{7} + \dfrac{4y}{7} &= 10 \\ \dfrac{x}{2} + \dfrac{y}{3} &= 5 \end{aligned}
c
\begin{aligned} -\dfrac{6 x}{5} + \dfrac{3 y}{5} &= \dfrac{6}{5} \\ - \dfrac{1}{4} \left( - 5 x + \dfrac{y}{3}\right) &= 2 \end{aligned}
17

Use the elimination method to solve the following pairs of equations:

a
\begin{aligned} 2 x + 5 y &= 44 \\ 6 x - 5 y &= - 28 \end{aligned}
b
\begin{aligned} 8 x + 3 y &= - 11 \\ - 8 x - 5 y &= 29 \end{aligned}
c
\begin{aligned} 2 x - 5 y &= 1 \\ - 3 x - 5 y &= - 39 \end{aligned}
d
\begin{aligned} 7 x - 4 y &= 15 \\ 7 x + 5 y &= 60 \end{aligned}
e
\begin{aligned} - 6 x - 2 y &= 46 \\ - 30 x - 6 y &= 246 \end{aligned}
f
\begin{aligned} - 5 x + 16 y &= 82 \\ 25 x - 4 y &= 122 \end{aligned}
g
\begin{aligned} 0.2 x + 0.3 y &= 0.5 \\ 0.5 x + 0.4 y &= 0.2 \end{aligned}
CAS calculator
18

For each of the following system of equations:

i

Graph the two linear functions using the graph mode of your CAS calculator and state how many solutions are there to the system of linear equations.

ii

How could the answer to part (a) have been predicted from looking at the equations?

a
\begin{aligned} 6 x - y &= 1 \\ 12 x - 2 y &= 2 \end{aligned}
b
\begin{aligned} - 9 x + 8 y &= - 5 \\ 5 x + 7 y &= - 9 \end{aligned}
c
\begin{aligned} - 3 x + y &= - 5 \\ 9 x - 3 y &= - 4 \end{aligned}
19

Use your CAS calculator to graph the following systems of equations, and hence state the values of x and y which satisfy both equations:

a
\begin{aligned} 4 x - 3 y &= - 13 \\ 4 x + 3 y &= 5 \end{aligned}
b
\begin{aligned} y &= 4 x \\ y &= 27 - 5 x \end{aligned}
c
\begin{aligned} \dfrac{x}{3} + \dfrac{2y}{3} &= 1 \\ \dfrac{x}{2} + \dfrac{y}{3} &= 7 \end{aligned}
d
\begin{aligned} 3 x - 1.6 y &= 5.9 \\ 0.45 x + 0.7 y &= - 0.02 \end{aligned}
20

Consider the following equations:

  • Equation 1: x - y = - 6

  • Equation 2: - x + 2 y = 9

  • Equation 3: 2 x - 7 y = - 42

a

Graph the lines for equations 1 and 2 using the graphing functionality of your CAS calculator and state the solution to the system of this pair of equations.

b

Graph the lines for equations 1 and 3 using the graphing functionality of your CAS calculator and state the solution to the system of this pair of equations.

c

Graph the lines for equations 2 and 3 using the graphing functionality of your CAS calculator and state the solution to the system of this pair of equations.

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U1.AoS4.8

solve linear equations constructed from word problems, including simultaneous linear equations

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