Topics
1. Simultaneous linear equations
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AustraliaNSW
Year 12

1.03 Solving simultaneous equations using technology

Lesson
Worksheet
Practice
Worksheet
Simultaneous equations with technology
1

For each of the following systems of equations, graph both equations using technology and state the values of x and y that satisfy both equations:

a
\begin{aligned} y &= x \\ y &= -4 - 3x \end{aligned}
b
\begin{aligned} y &= 2x \\ y &= -42 - 5x \end{aligned}
c
\begin{aligned} y &= 4x \\ y &= -7 + 3x \end{aligned}
d
\begin{aligned} y &= - 2 x \\ y &= 4 - 4 x \end{aligned}
e
\begin{aligned} y &= - 3 x \\ y &= 10 - 5 x \end{aligned}
f
\begin{aligned} 4 x - 3 y &= - 22 \\ - 3 x - 2 y &= 8 \end{aligned}
2

For each of the following systems of equations, graph both equations using technology and state the point of intersection:

a
\begin{aligned} y &= 2 x \\ y &= - 35 - 3 x \end{aligned}
b
\begin{aligned} y &= 5x + 23 \\ y &= 2x + 8 \end{aligned}
c
\begin{aligned} y &= 6 x + 30 \\ y &= 4 x + 22 \end{aligned}
d
\begin{aligned} y &= 2 x - 5 \\ 5 x - y &= 17 \end{aligned}
e
\begin{aligned} y &= 2x - 1 \\ 5x - y &= 7 \end{aligned}
f
\begin{aligned} - 2 x + 5 y &= 19 \\ 3 x + 4 y &= 6 \end{aligned}
g
\begin{aligned} -6x + 5y &= 28 \\ 2x + 3y &= 0 \end{aligned}
h
\begin{aligned} -4x + 3y &= -10 \\ -4x - 3y &= 2 \end{aligned}
i
\begin{aligned} y &= 3.5x - 16.78 \\ y &= -8.5x + 25.22 \end{aligned}
j
\begin{aligned} \dfrac{1}{5} x + \dfrac{3}{5} y &= 3 \\ \dfrac{2}{3} x + \dfrac{1}{4} y &= 4 \end{aligned}
3

For each of the following systems of equations, solve the system using technology, and state the values of x and y that satisfy both equations. Round your answers correct to two decimal places where necessary.

a
\begin{aligned} y &= 3.1x - 16.13 \\ y &= -8.1x + 11.87 \end{aligned}
b
\begin{aligned} y &= 3.2 x - 17.23 \\ y &= - 8.1 x + 22.32 \end{aligned}
c
\begin{aligned} y &= 3.8 x - 15.55 \\ y &= - 8.1 x + 38 \end{aligned}
d
\begin{aligned} 3x - 1.5y &= 4.3 \\ 0.95x + 0.2y &= -0.08 \end{aligned}
e
\begin{aligned} 3 x - 1.8 y &= 4.7 \\ 0.55 x + 0.6 y &= - 0.04 \end{aligned}
f
\begin{aligned} \dfrac{1}{5} x + \dfrac{3}{5} y &= 9 \\ \dfrac{1}{2} x + \dfrac{1}{3} y &= 7 \end{aligned}
g
\begin{aligned} \dfrac{1}{5} x + \dfrac{1}{5} y &= 4 \\ \dfrac{1}{2} x + \dfrac{1}{3} y &= 3 \end{aligned}
h
\begin{aligned} \dfrac{1}{3} x + \dfrac{1}{3} y &= 7 \\ \dfrac{1}{2} x + \dfrac{1}{3} y &= 3 \end{aligned}
4

Solve the following pairs of equations by using technology, to graph the lines on the same number plane:

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

Graph the following lines on the same set of axes using technology, to determine how many solutions there are to the system of equations:

\begin{aligned} 6x - y &= 1 \\ 12x - 2y &= 2 \end{aligned}
6

Solve the following systems of equations using technology:

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} \dfrac{x}{2} + y &= 3 \\ \dfrac{x}{5} + 3 y &= - 4 \end{aligned}

h

\begin{aligned}x + \dfrac{5}{4} y &= \dfrac{9}{4} \\ \dfrac{3}{5} x + y &= \dfrac{7}{5} \end{aligned}

i

\begin{aligned} - \dfrac{x}{4} + \dfrac{y}{5} &= 8 \\ \dfrac{x}{5} + \frac{y}{3} &= 1 \end{aligned}

j

\begin{aligned} 0.4 x - 0.63 y &= 0.23 \\ 2 x + 7 y &= - 9 \end{aligned}

k

\begin{aligned} 5 x + 3 y &= 7 \\ x + y &= 2 \end{aligned}

l

\begin{aligned} - 5 p - 7 q &= - \dfrac{43}{5} \\ -18p - 28q &= - \dfrac{187}{5} \end{aligned}

7

Consider the following equations:

  • Equation 1: x - y = - 6

  • Equation 2: - x + 2 y = 9

  • Equation 3: 2 x - 7 y = - 42

Graph the following pairs of equations using technology, and state the solution to each system of equations:

a

Equations 1 and 2

b

Equations 1 and 3

c

Equations 2 and 3

Applications
8

Consider the following system of linear equations:

\begin{aligned} - 6 x - 2 y &= - 28 \\ 2 x + 16 y &= 40 \\ 4 x - 2 y &= 12 \end{aligned}
a

Find the values of x and y that satisfy the first two equations.

b

Determine if this solution satisfies the third equation by substituting the values of x and y into the left hand side of the equation.

c

Hence state whether the lines are concurrent.

9

The percentage of the workforce (y) that are teenagers is modelled by 3.3 x + y = 35.3. The percentage of the workforce that are pensioners is modelled by 3.2 x - y = - 28.8, where x is the number of years since 2018.

a

Use technology to find the x and y values that satisfy both equations.

b

State the year in which the proportion of the workforce that are teenagers and the proportion of the workforce that are pensioners is the same.

c

State the percentage of the workforce that are teenagers (or the percentage of the workforce that are pensioners) in this year.

10

Consider the straight line y = a x + b that passes through the two points \left(5, 3\right) and \left(8, 0\right).

a

Write a pair of simultaneous equations using the points given.

b

Find the value of a and b.

c

State the equation of the straight line that passes through the points \left(5, 3\right) and \left(8, 0\right).

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Outcomes

MS2-12-1

uses detailed algebraic and graphical techniques to critically evaluate and construct arguments in a range of familiar and unfamiliar contexts

MS2-12-6

solves problems by representing the relationships between changing quantities in algebraic and graphical forms

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