Year 8: Changing the Subject

Year 8: Changing the Subject

Dr J Frost ([email protected])www.drfrostmaths.com

Last modified: 8th February 2016

𝑥=10

Starter

Solve the following equations:

2𝑥+73 =9 ?

3 𝑥2+5=152 𝑥=7?

S1

S2

S3

5 𝑥−2=3 𝑥+3 𝑥=52?

Solving Linear EquationsImagine the stuck inside a prison – we gradually have to ‘undo’ the things around it before it can be released. Undo the last thing done to on each step by doing the opposite.

√2 𝑥−5=3Click We can’t add 5 yet because

it’s ‘trapped’ inside .We ‘square’ to undo the .

2 𝑥−5=9Click

2 𝑥=14Click

𝑥=7

Test Your Understanding

1 Solve Solve 2

Solve 3

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Changing the Subject of a Formula

The formula to calculate a temperature in Fahrenheit if we have the temperature in Celsius:

But what if we had say the temperature in Fahrenheit, and wanted to know it in Celsius?

𝐶=59

(𝐹−32 )The subject of the formula is the variable the appears on its own on one side of the equation (usually the left) and not on the other side.

Skill #1: ‘Undoing’ to UnlockMake the subject of the formula. Undo the last thing done to the subject each time

by doing the opposite.

𝑦=𝑥−2 𝑥=𝑦+2

𝑦=3 𝑥+2 𝑥=𝑦−23

𝑦=√𝑥+1 𝑥=(𝑦−1 )2

𝑦=𝑥2−𝑎4

𝑥=±√4 𝑦+𝑎

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Bro Tip: It doesn’t matter what side the subject is on, provided it’s on its own!

Exercise 1In each case make the subject of the formula. (Please copy out question first)

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?????

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14

RED ORANGE GREEN BLUE

Vote with your diaries

Test your understanding so far…

𝑥=𝑦+32𝑥=

𝑦+32 𝑥=2 𝑦−3𝑥=

𝑦−32

𝑦=2 𝑥−3Make the subject.

𝑥=𝑦2𝑎

𝑥=( 𝑦𝑎 )2

𝑥=√𝑦𝑎 𝑥=(𝑎𝑦 )2

𝑦=𝑎√𝑥Make the subject.

𝒙=±√𝒚 −𝟏𝒂 𝒙=±√ 𝒚𝒂 −𝟏𝒙=±√ 𝒚−𝟏𝒂 𝒙=±√ 𝒚𝒂 −𝟏𝑦=𝑎𝑥2+1Make the subject.

𝒙=( 𝒚𝒃 )𝟐−𝟏𝒙=

(𝒚 −𝟏 )𝟐

𝒃 𝒙=𝒚𝟐

𝒃 −𝟏𝒙=𝒚𝟐−𝟏𝒃

𝑦=𝑏√𝑥+1Make the subject.

Skill #2: Subject trapped in a negative term

When the subject is within the first argument of a subtraction, it’s easy to ‘release’.

𝑦=2 𝒙−3 2 𝒙=𝑦 +3?

However, it’s a tiny bit harder if the subject is in the term being subtracted.

𝑦=3−2𝑥 𝑦+2𝑥=32 𝑥=3− 𝑦

When the subject is inside a negative term, just add it to both sides.

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Examples

𝑎− 𝑥=𝑏 1−𝑏𝑥=𝑐1 2

3

𝑎𝑏−𝑐 √𝑥=𝑦+1

??

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Doing it in one step… (if you like)

How could you rearrange the numbers in to get another subtraction?

This suggests you can swap the thing you’re subtracting with the result. (i.e. Only the thing to the left of the subtraction stays put)

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Examples:

𝑎− 𝑥=𝑏??

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56

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???????

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RED ORANGE GREEN BLUE

Vote with your diaries

Test your understanding so far…

x = y + 3 x = y – 3 x = 3 – y x = 3y

𝑦=3−𝑥Make the subject.

𝑥=3

𝑦−1 𝑥=3𝑦 −1𝑥=

𝑦3 −1𝑥=

𝑦2

𝑦=3𝑥+1

Make the subject.

𝑥=𝑦2 − 𝑧𝑥=𝑦−2𝑧𝑥=

𝑦−𝑧2

𝑦=2 (𝑥+𝑧 )Make the subject.

𝑥=𝑦−12 𝑥=

1+𝑦2 𝑥=1−2 𝑦𝑥=

1− 𝑦2

𝑦=1−2 𝑥Make the subject.

𝑥=𝑞𝑦 −1

Skill #3: Subject trapped in a denominatorWhen the subject is in the numerator of a fraction, it’s easy to ‘release’ the subject from the fraction.

𝑦=𝑥𝑞 𝑥=𝑞𝑦?

But it’s a bit harder if the subject is in the denominator…

𝑦=𝑞𝑥+1 𝑦 (𝑥+1 )=𝑞

𝑥+1=𝑞𝑦

In general, whenever you have a fraction in an equation, your instinct should be to multiply both sides by the denominator.

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Skill #3: Subject trapped in a denominator

! Isolate the fraction on one side of the equation, then multiply by denominator.

𝑎=𝑏−𝑐𝑥

𝑎𝑥 +𝑏=𝑐

𝑎=𝑏− 𝑐𝑥

1 2

3

??

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+2 first as was last thing done to

Doing it in one step… (if you like)

How would you rearrange the numbers in to get another division?

Another way of thinking about it… Remember your speed-distance-time triangle from physics? Once you get to a point where you have to release from the fraction, you can apply what I call the ‘triangle trick’.

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Examples:

𝑐=𝑏

4−2 𝑥 4−2 𝑥=𝑏𝑐

𝑎=𝑏𝑥

𝑎−𝑏2𝑥+1

=𝑐 𝑦=𝑦 2𝑥+ 𝑦 −2

E1 E2E3

? ? ?

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Thus we can swap the thing we’re dividing by and the result. The numerator is left unchanged.

Skill #3b: ‘Cross multiplying’

If you have just a fraction on each side of the equation, you can ‘cross multiply’.

𝑎𝑏

𝑐𝑑¿ Click for

Bromanimation

Examples:Make the subject:

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E1E2


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Questions similar to those in Tiffin CATsMake the subject of the formula:

or ?

Make the subject of the formula:

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𝑥=𝑦

𝑎−1 𝑥=𝑎𝑦 −1𝑥=

𝑎𝑦−1𝑥=2

𝑦=𝑎𝑥+1

Make the subject.

𝑥=𝑦2 +1𝑥=2 𝑦+1𝑥=𝑦+2𝑥=2 𝑦+2

𝑦=𝑥2 −1

Make the subject.

𝑥=±√ 𝑎𝑦 𝑥=±√ 𝑦𝑎 𝑥=𝑦±√𝑎𝑥=±√𝑦𝑎

𝑦=𝑎𝑥2Make the subject.

𝑥=±√ 2𝑦𝑥=2𝑦 𝑥=±√ 𝑦2𝑥=±√2 𝑦

𝑦=2𝑥2

Make the subject.

𝑥=( 𝑦𝑎+1)2

𝑥=𝑦2−1𝑎 𝑥=

(𝑦+1 )2

𝑎 𝑥=𝑦2+1𝑎

𝑦=𝑎√𝑥−1Make the subject.

𝑥=1

(𝑦−1 )2−1 𝑥=

11− (1− 𝑦 )2𝑥=𝑦 2−1

𝑦=1

√𝑥+1+1

Make the subject.

𝑥=( 𝑏𝑐𝑎−1 )2𝑥=𝑎𝑏𝑐𝑥=( 𝑎𝑐𝑏+1 )

2

𝑥=(𝑎 (𝑏+1 )𝑐 )

2

𝑎√𝑥

=𝑏+1𝑐

Make the subject.

Activity Time!

There are 3 levels, each of increasing difficulty.

Once you’ve completed all the questions in Level 1, check your answers with me – after which you can advance onto the next level.

Merit on offer to anyone who can complete all Level 3 questions.

Substituting

𝒚=𝟒 ,𝒂=𝟐𝟎Bro Tip: Write out the values of your variables first using the information given.

𝒂=𝟑𝟎 ,𝒄=𝟏𝟓When we substitute our numbers in, we now have to to rearrange to solve!

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Q

Test Your UnderstandingThe maths exam mark of an 8EWS student is determined by the hours revised and the number of cats they have using the following formula:

Given that Jaimal has 2 cats and gets a mark of 80, how many hours did he revise?

Q QThe ‘cool coefficient’ of a boy is determined by their number of skateboards and the mass of hair gel they apply, using the formula:

a) If Max has a Cool Coefficient of 5 and has 10 skateboards, how many kilograms of hair gel does he use?

b) If he has a Cool Coefficient of 4 and uses 2kg of hair gel, how many skateboards does he have?

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Exercise 4

Given that and and , find

Given that and , find

a) Find when

b) Find when

The price of a car with initial price and age is given by the formula:

a) Find the current price of the car when it is initially £10 000 and is 5 years old.

b) What is its age when a car initially worth £10 000 has fallen to a tenth of its value?

You can calculate a temperature in Fahrenheit from Celsius using the formula:

It is currently . What is the temperature in Fahrenheit?86F

Determine when

Find when

Given that the conversion between Fahrenheit and Celsius is:

Determine the temperature which is the same in both Fahrenheit and Celsius.

Solving:

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Documents

Year 8: Changing the Subject