AC5 Simultaneous Equations (one non-linear)

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From users who are members of Skills Audits for Maths and Stats :

Lauren Desoysa	said	Ready to use	1 year, 1 month ago
Andrew Neate	said	Needs to be tested	1 year, 2 months ago

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Will Morgan 1 year ago

Published this.

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Lauren Desoysa 1 year, 1 month ago

Gave some feedback: Ready to use

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Andrew Neate 1 year, 2 months ago

Gave some feedback: Needs to be tested

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Andrew Neate 1 year, 2 months ago

Created this as a copy of Simultaneous Equations : Quadratics 2.

Name	Status	Author	Last Modified
Simultaneous Equations : Quadratics 1	Ready to use	Ben McGovern	06/09/2023 13:28
Simultaneous Equations : Quadratics 2	Ready to use	Ben McGovern	14/09/2022 17:27
Simultaneous Equations : Quadratics 3	Ready to use	Ben McGovern	06/09/2023 13:23
Simultaneous Equations : Quadratics 4	Ready to use	Ben McGovern	14/09/2022 17:28
AC5 Simultaneous Equations (one non-linear)	Ready to use	Andrew Neate	09/04/2024 15:11

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Question statement

Solve the following simultaneous equations:

$\begin{split} \simplify{{a1}x+y} &\,= \var{c1} \\ \simplify{{a2}x^2+{b2}x*y} &\,= \var{c2} \end{split}$

Give your answers to 2 decimal places where necessary.

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	-3
c1	integer	1
a2	integer	2
b2	integer	1
c2	integer	7
solx1	number	-1.2874342087
solx2	number	1.0874342087
soly1	number	-2.8623026261
soly2	number	4.2623026261
x1dp	number	-1.29
y1dp	number	-2.86
x2dp	number	1.09
y2dp	number	4.26
solutions1	matrix	matrix([-1.29,-2.86])
solutions2	matrix	matrix([1.09,4.26])
check	string	<p>\[ x_1 = \var{x1dp} \,&nbsp
text	string	<p>\[ x_1 = \var{x1dp} \,&nbsp
text1	string	<p>\[ x_1 = \var{x1dp} \,&nbsp
text2	string	<p>\[ x_1 = \var{x1dp} \, \tex
text3	string	<p>\[ x_1 = \var{x1dp} \,&nbsp

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-5..-1)

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

-3

→ Used by:

solx1
solx2
soly1
soly2

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Matrix entry
  1. Alt Unnamed alternative
  Add
2. Gap Gap 1. Matrix entry
  1. Alt Unnamed alternative
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$(x_1,y_1)=$ Gap 0.

$(x_2,y_2)=$ Gap 1.

Advice

To solve a pair of simultaneous equations of this type we want to rearrange the linear equation such that $y$ is the subject, which we can then substitute into the equation with the quadratic $x$ -term. This will result in a quadratic equation in terms of $x$ only.

For the equations

$\begin{split} \simplify{{a1}x+y} &\,= \var{c1} \qquad \qquad &(1) \\\simplify{{a2}x^2+{b2}x*y} &\,= \var{c2} \qquad \qquad &(2) \end{split}$

we can rearrange equation (1) to make $y$ the subject:

$y = \simplify{{c1}-{a1}x}. \qquad\qquad (3)$

Substituting this into equation (2):

$\begin{split}\simplify{{a2}x^2+{b2}x({c1}-{a1}x)} &\,=\var{c2} \\ \simplify[!cancelTerms,unitFactor]{{a2}x^2+{b2*c1}x-{b2*a1}x^2} &\,=\var{c2}. \end{split}$

Collecting similar terms:

$\simplify{({a2}-{b2*a1})x^2+{b2*c1}x-{c2}} =0. \qquad\qquad (4)$

Using the quadratic formula, we find two solutions for $x$ :

{check}

Therefore, the 2 pairs of solutions for these simultaneous equations are

$(x_1,y_1) = (\var{x1dp},\var{y1dp})$ and

$(x_2,y_2) = (\var{x2dp},\var{y2dp}).$

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Taxonomy: mathcentre

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Will Morgan 1 year ago

Lauren Desoysa 1 year, 1 month ago

Andrew Neate 1 year, 2 months ago

Andrew Neate 1 year, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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AC5 Simultaneous Equations (one non-linear)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 1 year ago

Lauren Desoysa 1 year, 1 month ago

Andrew Neate 1 year, 2 months ago

Andrew Neate 1 year, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`