Slideshow 12, Mathematics, Mr Richard Sasaki Substitution Slideshow 12, Mathematics, Mr Richard Sasaki

Objectives Review previous algebraic rules Be able to substitute numbers into expressions Be able to substitute into expressions with brackets

Algebraic Laws - Review −1×𝑥 = −𝑥 5𝑎+𝑎 = 6𝑎 𝑦×𝑦 = 𝑦 2 12𝑏−𝑏 = 11𝑏 𝑥×𝑦 = 𝑥𝑦 12𝑏÷𝑏 = 12 𝑥+𝑥 = 2𝑥 2 𝑥 5 × 𝑥 4 = 2𝑥 9 𝑥+𝑦 = 𝑥+𝑦 9 𝑦 7 ÷3 𝑦 3 = 3 𝑦 4 𝑥 ×2𝑥 = 2 𝑥 2 𝑥 3 4 = 𝑥 12 𝑎−𝑎 = 4𝑥 2 16 𝑥 2 𝑥÷𝑦 = 𝑥 𝑦 = 12𝑥÷𝑥 = 12 7 𝑥 7 𝑥 −1 = 2 𝑥 0 = 2 2(3𝑎+𝑏) = 6𝑎+ 2𝑏

Substitution We will now start giving values to unknowns. How can we represent 4 Five Yen coins and 2 One Thousand Yen notes algebraically? 4𝑥+2𝑦 Here, 𝑥 and 𝑦 refer to the value of these items. We know their values, right? 𝑥= 5 𝑦= 1000 So as 𝑥=5, 𝑦=1000, what is the value of 4𝑥+2𝑦? 4𝑥+2𝑦= 4×5+2×1000 =20+2000 =2020

Substitution This is substitution. We simply swap unknowns for numbers (their values). Example Calculate 𝑥+𝑦 when 𝑥=4 and 𝑦=7. 𝑥+𝑦= 4+7 =11 Easy, yeah? We just literally swap like that. Example Note: Remember, 3𝑥 means 3×𝑥. Calculate 3𝑥−2𝑦 when 𝑥=4 and 𝑦=3. 3𝑥−2𝑦= 3×4−2×3 =12−6 =6

Answers – Part 1 5 5 6 4 6 12 6 2 6 7 4 2 3 7 14 3 11 −6 −7 −21

Worded Examples As shown before, modelling worded problems makes things simpler. Example A man walks around a circuit three times (at a constant speed) and then runs around the circuit twice (at a different constant speed). Model: 3𝑥+2𝑦 =3×2+2×0.5 Note: 𝑥 and 𝑦 can represent a number of seconds, minutes, hours or any unit of time. He takes 2 minutes to walk a lap and 30 seconds to run one. How long does he take in total? 7 minutes.

Answers – Part 2 3𝑥+2𝑦 720 𝑌𝑒𝑛 730 𝑌𝑒𝑛 810 𝑌𝑒𝑛 Aizawa 34𝑥+72𝑦 140 𝑐𝑚 < > 352 𝑐𝑚 450 𝑐𝑚 > 2𝑘𝑔 2000𝑔 ⇒ equal 1850𝑔⇒ lighter

Order of Operations We must review the order that we calculate in. Example Calculate 4 2 +3×(2−1). = 4 2 +3× 1 =16+3× 1 =16+3 =19 Remember, calculate things in brackets first, then powers, then division / multiplication and lastly addition / subtraction. Note: Division is another style of multiplication. Addition is another style of subtraction.

8 2 10 3 14 64 7 9 6 6 80 2 14 4 1 2 19 81 16 25

Non-Linear Substitution We saw the word linear in sequences. In algebra, linear means that no unknowns multiply other unknowns or themselves. Yes. Is 3𝑥−4 linear? Yes. Is 2𝑥 linear? Is 𝑥𝑦 linear? No. Is 𝑥 2 linear? No. Non-linear expressions may include things like 𝑥𝑦 and 𝑥 2 . Examples Calculate 3 𝑥 2 +𝑦 when 𝑥=2, 𝑦=0. 3 𝑥 2 +𝑦= 3× 2 2 +0 =3×4+0 =12 Calculate 2𝑥𝑦+4 when 𝑥=4, 𝑦=3. Note: Remember this! 2𝑥𝑦+4= 2×4×3+4 =24+4 =28

4 1 4 24 50 2 49 21 42 189 3 20 −1 −9 7 −2 4 17 2 1 38 136

Brackets and Substitution We have only looked at brackets in simple cases. We can expand first and then substitute or substitute immediately. Example Calculate 4 𝑥 2 −𝑦 when 𝑥=5, 𝑦=2. 4( 𝑥 2 −𝑦)= 4 𝑥 2 −4𝑦= 4× 5 2 −4×2 =100−8 =92 We can of course substitute first. 4( 𝑥 2 −𝑦)= 4( 5 2 −2)= 4×23 =92 Note: You should be able to substitute using both of these methods.

2𝑥+6 12 4𝑥+8𝑦 28 4𝑥−𝑥𝑦 6 2 𝑥 2 −2𝑦 14 5𝑎−5𝑏 30 2−2𝑎𝑏 18 20−2 𝑏 2 12 2 𝑎 3 −2𝑎 𝑏 2 96 21 45 441