Simultaneous Equations : Quadratics 3

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Ben McGovern	said	Ready to use	3 years, 9 months ago
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Ben McGovern 2 years, 7 months ago

Published this.

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Ben McGovern 3 years, 9 months ago

Gave some feedback: Ready to use

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Ruth Hand 3 years, 10 months ago

Gave some feedback: Doesn't work

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Ruth Hand 3 years, 10 months ago

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Ruth Hand 4 years, 1 month ago

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Ben McGovern 4 years, 9 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:

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

Give your answers correct to 2 decimal places.

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	4
c1	integer	2
a2	integer	1
b2	integer	2
c2	integer	3
solx1	number	0.3819660113
solx2	number	2.6180339887
soly1	number	1.3090169944
soly2	number	0.1909830056
denom1	decimal	dec("1")
denom2	decimal	dec("2")
denom3	decimal	dec("3")
denom4	decimal	dec("4")
denom5	decimal	dec("5")
mult1	integer	1
mult2	integer	2
mult3	integer	3
mult4	integer	4
x1dp	number	0.38
x2dp	number	2.62
y1dp	number	1.31
y2dp	number	0.19
solutions1	matrix	matrix([0.38,1.31])
solutions2	matrix	matrix([2.62,0.19])

Automatically regenerate variables?

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Value

xxxxxxxxxx
 
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Description

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Generated value: `integer`

→ Used by:

denom1
denom2
denom3
denom4
denom5
solx1
solx2
soly1
soly2

This variable doesn't seem to be used anywhere.

Parts

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

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$(x_1,y_1)=$ Gap 0.

$(x_2,y_2)=$ Gap 1.

Advice

To solve a pair of simultaneous equations of this type we want to rearrange the first equation such that $y$ is the subject, which we can then substitute into the second equation. This can then be rearranged, resulting in a quadratic equation in terms of $x$ only.

For the equations

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

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

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

Substituting this into equation (2):

$\simplify{{a2}x+{b2}({c1}/({a1}x))} = \var{c2}.$

Now multiplying this equation by $\simplify{{mult1}x}$ and rearranging:

$\begin{split} \simplify{{a2*mult1}x^2+{b2*mult1}({c1}/({a1}))} &\,= \var{c2*mult1}x \\ \Leftrightarrow \qquad \simplify{{a2*mult1}x^2 - {c2*mult1}x+ {b2*mult1}({c1}/({a1}))} &\,= 0. \end{split}$

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

$x_1 = \var{x1dp} \, \text{(2 d.p.)} \quad \text{and} \quad x_2=\var{x2dp} \, \text{(2 d.p.)}$

To find the corresponding $y$ -values, we can plug these solutions for $x$ back into equation (3), which gives:

$y_1 = \var{y1dp} \, \text{(2 d.p.)} \quad \text{and} \quad y_2=\var{y2dp} \, \text{(2 d.p.)}$

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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Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ruth Hand 3 years, 10 months ago

Ruth Hand 3 years, 10 months ago

Ruth Hand 4 years, 1 month ago

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Simultaneous Equations : Quadratics 3

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ruth Hand 3 years, 10 months ago

Ruth Hand 3 years, 10 months ago

Ruth Hand 4 years, 1 month ago

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`