Substituting into Formulae

GCSE Maths Algebra Substituting into Formulae

In algebra, substitution is putting numbers in place of letters to calculate the value of an expression. For example:

If e=3 and f=4. what is that value of $40ef$ ?

40ef is the same as writing $40$ multiplied by $e$ multiplied by $f$ .

First of all, as we know that e=3, we replace e with 3 in the expression and we are left with: 40 multiplied by 3 multiplied by f.

Likewise, we do the same with f. We replace f with 4 in the expression and we are left with: 40 multiplied by 3 multiplied by 4.

Therefore, $40ef=40\times 3\times 4=280$ .

A formula is a mathematical relationship connecting two or more quantities. For example, $F = ma$ , expressing F in terms of m and a.

$X=\frac { -b+ \sqrt { { b }^{ 2 }-4ac } }{ 2a }$ is a formula giving x in terms of the quantities of a, b and c.

From F=ma, we can find the value of F if we are given values for m and a, which you can see in the example below.

Example

Find the value of F if $m = 3$ and $a = 14$

$F = ma$

$= 3 \times 14$

$= 42$

Example

From the formula $X=\frac { -b+ \sqrt { { b }^{ 2 }-4ac } }{ 2a }$ , if we are given that a = 2, b = 5 and c = −12, find the value of x.

We know that a=2, b=5 and c=−12, so:

$X=\frac { -5+ \sqrt { { 5 }^{ 2 }-4(2)(-12) } }{ 2(2) }$

$X=\frac { -5+\sqrt { 25+96 } }{ 4 }$

$=\frac { -5+\sqrt { 121 } }{ 4 }$

$=\frac { -5+11 }{ 4 }$

$=\frac { 6 }{ 4 }$

$=\frac { 3 }{ 2 }$

Example

Find the value of T when $m = 5$ , $g = 10$ and $w = -3$ , that:

$T = mg + mw(g^{ 2 } - 81)$ .

$T = mg + mw({ g }^{ 2 } - 81)$

Substituting $m = 5$ , $g = 10$ and $w = -3$ gives:

$5(10) + 5(-3) (10^{ 2 } - 81)$

$= 50 - 15(100 - 81)$

$= 50 - 15(19)$

$= 50 - 285$

$= -235$

Example

Find the value of S if:

$S = ut + \dfrac { 1 }{ 2 }{ at }^{ 2 }</mark>$

When $u = 3$ , $t = 5$ and $a = 4$ .

$S = ut + \dfrac { 1 }{ 2 }{ at }^{ 2 }$

$= (3) (5) + \dfrac{ 1 }{ 2 } (4) (5^{ 2 })$

$= 15 + 50$

$= 65$

Example

Given that ${ V }^{ 2 } = u^{ 2 } + 2aS$ . Find the possible values of V when u = 4, a = 5 and S = 4.8

Substituting $u = 4$ , $a = 5$ and $S = 4.8$ into the formula gives us:

$V^{ 2 } = 4^{ 2 } + 2(5) (4.8)$

${ V }^{ 2 } = 16 + 10 \times 4.8$

${ V }^{ 2 } = 16 + 48$

${ V }^{ 2 } = 64$

$V = \sqrt { 64 }$

$V = -8 or +8 = \pm8.$

Example

Find the possible values of x if $x = \sqrt {a^{ 2 }+b^{ 2 } }$ , when $a = 5$ and $b = 12$

So, we will use the formula $x = \sqrt { { a }^{ 2 }+{ b }^{ 2 } }$

Substituting $a = 5$ and $b = 12$ gives us:

$x = \sqrt { { 5 }^{ 2 }+{ 12 }^{ 2 } }$

$= \sqrt { 25+144 }$

$= \sqrt { 169 }$

$= \pm13$

Example

Find the value of w if $w = { u }^{ 2 }+{ v }^{ 2 }+{ z }^{ 2 }+{ a }^{ 3 }+{ b }^{ 3 }$ if:

$u = 1, v = 2, z = 3, a = 0, b=4$ .

We substitute u=1, v=-2, z=-3, a=0 and b=4 into the formula to get:

$w = { 1 }^{ 2 }+(-2)^{ 2 }+{-3}^{ 2 }+{ 0 }^{ 3 }+{ 4 }^{ 3 }$

$= 1 + 4 + 9 + 0 + 64$

$= 78$

Example

Evaluate y in each of the following expressions, given that:

$x = 2$ , $z = 3$ and $w = -1$

i) $y = { x }^{ z } + { z }^{ x } - w$

ii) $y = { z }^{ x } + w - { w }^{ 2 }$

iii) $y = \frac { { x }^{ 2 }-{ w }^{ 2 } }{ z }$

iv) $y = \frac { x+z+w }{ xzw }$

v) $y = \frac { xzw }{ x+z+w }$

i) $y = { x }^{ z } + { z }^{ x } - w$

$= { 2 }^{ 3 } + { 3 }^{ 2 } - (-1)$

$= 8 + 9 + 1$

$= 18$

ii) $y = { z }^{ x } + w - { w }^{ 2 }$

$= { 3 }^{ 2 } + (-1) - { (-1) }^{ 2 }$

$= 9 - 1 - 1$

$= 7$

iii) $y = \frac { { x }^{ z }-{ w }^{ z } }{ z }$

$= \frac { { 2 }^{ 2 }-{ (-1) }^{ 2 } }{ 3 }$

$=\frac{4-1}{3}$

$= \frac{3}{3}$

$=1$

iv) y = \frac{x+z+w}{xzw}

= \frac{2+3-1}{2(3)(-1)}

= \frac{4}{-6}

= \frac{-2}{3}

v) $y = \frac {xzw}{x+z+w}$

*** QuickLaTeX cannot compile formula:
<span style="color: #000000;">= \frac{2(3)(-1)}{2+3-1}

*** Error message:
You can't use `macro parameter character #' in math mode.
leading text: $<span style="color: #

*** QuickLaTeX cannot compile formula:
<span style="color: #000000;">= \frac{-6}{4}

*** Error message:
You can't use `macro parameter character #' in math mode.
leading text: $<span style="color: #

$= \frac{-3}{2}$

Example

Find the value of w when:

$x = -2$ , $y = -3$ , $x = \frac { 1 }{ 4 }$ and $w = { (\frac { xy }{ z } +3) }^{ \frac { 1 }{ 3 } }$

Substituting $x = -2$ , $y = -3$ and $z = \frac { 1 }{ 4 }$ into the formula gives:

$w = { (\frac { (-2) (-3) }{1/4} +3) }^{ \frac { 1 }{ 3 } }$

$= { (\frac { 6 }{ 1/4 } +3) }^{ \frac { 1 }{ 3 } }$

$= { (24+3) }^{ \frac { 1 }{ 3 } }$

$= { 27 }^{ \frac { 1 }{ 3 } }$

$= { { (3 }^{ 3 }) }^{ \frac { 1 }{ 3 } }$

$= 3$

Example

Giving that $E = { mc }^{ 2 }$ . Find the value of E when $m = 100,000,000$ and $c = 300,000,000$ , giving an answer in standard form.

Substituting $m = { 10 }^{ 8 }$ , $c = 3 \times { 10 }^{ 8 }$ in $E = { mc }^{ 2 }$ gives:

$E = { 10 }^{ 8 } \times { (3\times { 10 }^{ 8 }) }^{ 2 }$

*** QuickLaTeX cannot compile formula:
= { 10 }^{ 8 } \times { 3 }^{ 2 } \times ({ 10 }^{ 8 }) }^{ 2 }

*** Error message:
Extra }, or forgotten $.
leading text: ... \times { 3 }^{ 2 } \times ({ 10 }^{ 8 }) }

$= { 10 }^{ 8 } \times { 3 }^{ 2 } \times { 10 }^{ 16 }$

$= { 3 }^{ 2 } \times { 10 }^{ 8 + 16 }$

$= 9 \times { 10 }^{ 24 }$

Example

Find the value of x if $y=11115$ and $w=294$ , given that $x = y^{2} - w^{2}$

We substitute $y=11115$ and $w=294$ into the formula, which gives us:

$x = 11115^{2} - 294^{2}$

$= (11115+294)(11115-294)$

$=11409 \times 10821$

$= 123,456,789$

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Last updated: April 15, 2022

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Behavioural Remarketing

We use remarketing services to advertise to you after you have accessed or visited our website. We and our third-party vendors use cookies and non-cookie technologies to help us recognize your device and understand how you use our website. Therefore, we can improve our services to reflect your interests and serve you advertisements that are likely to be of more interest to you.

These third-party vendors collect, store, use, process and transfer information about your activity on our website in accordance with their Privacy Policies, which enables us to:

Measure and analyse traffic and browsing activity on our website
Show advertisements for our products and/or services to you on third-party websites or apps
Measure and analyse the performance of our advertising campaigns
Some of these third-party vendors may use non-cookie technologies that may not be impacted by browser settings that block cookies. Your browser may not permit you to block such technologies. Instead, you can use the following third-party tools to decline the collection and use of information for the purpose of serving you interest-based advertising:

The NAI’s opt-out platform: https://www.networkadvertising.org/choices/
The EDAA’s opt-out platform https://www.youronlinechoices.com/
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You may opt-out of all personalised advertising by enabling privacy features on your mobile device, such as Limit Ad Tracking (iOS) and Opt Out of Ads Personalisation (Android). Check your mobile device Help system for more information.

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The third-party vendors We use are:

• Google Ads (AdWords)
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You can opt-out of Google Analytics for display advertising and customise the Google Display Network ads by visiting the Google Ads Settings page: https://www.google.com/settings/ads

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• Facebook
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For more information on the privacy practices of Facebook, please visit Facebook’s Data Policy: https://www.facebook.com/privacy/explanation

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We may use third-party Service Providers to provide a better improvement of our website. For example, we use:

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GDPR Privacy

Legal Basis for Processing Personal Data under GDPR

We may process personal data under the following conditions:

• Consent: You have given Your consent for processing personal data for one or more specific purposes.

• Performance of a contract: The provision of personal data is necessary for the performance of an agreement with you or for any pre-contractual obligations thereof.

• Legal obligations: Processing personal data is necessary for compliance with a legal obligation to which the company is subject.

• Vital interests: Processing personal data is necessary in order to protect your vital interests or of another natural person.

• Public interests: Processing personal data is related to a task that is carried out in the public interest or in the exercise of official authority vested in the Company.

• Legitimate interests: Processing personal data is necessary for the purposes of the legitimate interests pursued by the Company.

In any case, we will gladly help to clarify the specific legal basis that applies to the processing, and in particular whether the provision of personal data is a statutory (required by law) or a contractual requirement, or a requirement necessary to enter into a contract.

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We respect the confidentiality of your personal data and we guarantee that you can exercise your rights.

You have the right under this privacy policy, and by law if you are within the EU or the UK, to:

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You may exercise Your rights of access, rectification, cancellation and opposition by contacting us. Please note that we may ask you to verify your identity before responding to such requests. If you make a request, we will try our best to respond to you as soon as possible.

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Our website may contain content appropriate for children under the age of 13. As a parent, you should know that by using our services, children under the age of 13 may participate in activities that involve the collection or use of personal information. We use reasonable efforts to ensure that before we collect any personal information from a child, the child’s parent receives notice of and consents to our personal information practices.

We may also limit how we collect, use, and store some of the information of users between 13 and 18 years old. In some cases, this means that we will be unable to provide certain functionalities of our services to these users. If we need to rely on consent as a legal basis for processing your information and your country requires consent from a parent, We may require the parent’s consent before we collect and use that information.

We may ask a user to verify their date of birth before collecting any personal information from them. If the user is under the age of 13, the service will either be blocked or redirected to a parental consent process.

Information Collected from Children Under the Age of 13

We may collect and store persistent identifiers, such as cookies or IP addresses from children, without parental consent for the purpose of supporting the internal operation of the our services.

We may collect and store other personal information about children if this information is submitted by a child with prior parental consent or by the parent or guardian of the child.

We may collect and store the following types of personal information about a child when submitted by a child with prior parental consent or by the parent or guardian of the child:

First and/or last name
Date of birth
Gender
Grade level/year group
Email address
Telephone number
Parent’s or guardian’s name
Parent’s or guardian’s email address
For further details on the information we might collect, you can refer to the “Types of Data Collected” section of this privacy policy. We follow our standard privacy policy for the disclosure of personal information collected from and about children.

Parental Access

A parent who has already given the Company permission to collect and use their child’s personal information can, at any time:

Review, correct or delete the child’s personal information
Discontinue further collection or use of the child’s personal information
To make such a request, you can write to us using the contact information provided in this privacy policy.

Links to Other Websites

Our Service may contain links to other websites that are not operated by us. If you click on a third party link, you will be directed to that third party’s site. We strongly advise you to review the privacy policy of every site you visit.

We have no control over and assume no responsibility for the content, privacy policies or practices of any third party sites or services.

Policies Unique to Our Tutoring Services

The following applies to students that have registered for our tutoring services and tutors and employees representing Shalom Education Ltd:

Your profile includes information such as your username and password, purchases or orders made by you, your interests, preferences, and feedback and survey responses.

Booking history including booking requests, subjects booked, set-up test while using our digital resources, any online meeting and recordings of your bookings

Electronic recordings from CCTV, tutoring session, video conference calls, communication history with other users, your interaction with Shalom Education Tutors, records of referral codes shared, redeemed and any promotions that have been applied to your account. All photo’s, videos and conference recordings may be used to protect learner’s, Tutors and employees and may be used as evidence to support any complaint.

Changes to this Privacy Policy

We may update Our Privacy Policy from time to time. We will notify You of any changes by posting the new Privacy Policy on this page.

We may let You know via email and/or a prominent notice on our website prior to the change becoming effective and update the “Last updated” date at the top of this Privacy Policy.

You are advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page.

Contact Us

If you have any questions about this Privacy Policy, You can contact us:

By email: info@shalom-education.com
By visiting this page on our website: https://www.shalom-education.com/contact
By phone number: 01206657616

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