Simultaneous Equations : Quadratics 1

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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:

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

Give your answers to 2 decimal places where necessary.

Give any introductory information the student needs.

			Name	Type	Generated Value

a1	integer	-1
c1	integer	8
a2	integer	1
b2	integer	1
c2	integer	6
solx1	number	-5.4142135624
solx2	number	-2.5857864376
soly1	number	2.5857864376
soly2	number	5.4142135624
x1dp	number	-5.41
x2dp	number	-2.59
y1dp	number	2.59
y2dp	number	5.41
solutions1	matrix	matrix([-5.41,2.59])
solutions2	matrix	matrix([-2.59,5.41])

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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`

-1

This variable doesn't seem to be used anywhere.

Parts

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1. Gap Gap 0. Matrix entry
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2. Gap Gap 1. Matrix entry
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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 linear equation such that it is in terms of $x$ or $y$ , which we can then substitute into the equation with the quadratic terms. This will result in a quadratic equation in terms of one variable only.

For the equations

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

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

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

Substituting this into equation (2):

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

Collecting similar terms:

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

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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Simultaneous Equations : Quadratics 1

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

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`