Expanding brackets and substitution Slideshow 8, Room 307 Mathematics, Mr Richard Sasaki

Objectives Recall how to substitute into expressions Practice expanding brackets Expand brackets before substituting into expressions

Substitution To substitute, we swap algebraic symbols for numbers so we can get a numeric answer. Example Calculate 𝑥+𝑦 when 𝑥=4 and 𝑦=7. Try the short worksheet. 𝑥+𝑦 = 4+7 = 11 Usual algebraic rules for expressions apply.

Substitution Answers 9 2 56 144 75 32 8.5 𝟕𝒃 + 𝟒𝒄; 𝒃 = 𝟓𝟎, 𝒄 = 𝟏𝟐𝟎 67 830 Yen 12 4 𝒙 = 𝒎 + 𝒏 + 𝒅 + 𝒆 + 𝒇 = (-3) + 2 + 𝟒 𝟗 + (-2) + (-27) + (-2.5) = -32 – 𝟗 𝟏𝟖 + 𝟖 𝟏𝟖 = -32 𝟏 𝟏𝟖 8

Expanding brackets To substitute numbers into expressions with brackets, it can be easier to expand them first. This lesson, we will expand each expression before substituting.

= 32 + 24 = 56 = 4(2𝑥 + 3𝑦) 8𝑥 + 12𝑦 Expanding brackets Example Simplify 4(2𝑥 + 3𝑦) and substitute 𝑥 = 4 and 𝑦 = 2 into it. 4(2𝑥 + 3𝑦) = 8𝑥 + 12𝑦 = 32 + 24 That’s about it, try the last worksheets! 56 =

Answers 18 36 6 𝟏𝟐𝒂𝒃𝟐𝒄 72 𝟑(𝟑𝒄 + 𝟐𝒃) 1 𝟓𝟕𝟎𝟎 𝒀𝒆𝒏 𝟒(𝟑𝒙 − 𝒚𝟐) 510 𝟑𝒂+𝟑𝒃 18 𝟖𝒂 – 𝟒𝒃 36 𝟑𝒂𝒃 – 𝒂𝒃𝟐 6 𝟏𝟐𝒂𝒃𝟐𝒄 72 𝟑(𝟑𝒄 + 𝟐𝒃) 𝟑𝒙𝟐 𝟒𝒚 1 𝟑 𝟐 𝟑𝒙𝟐 𝟒𝒙 + 𝟖𝒚 𝟓𝟕𝟎𝟎 𝒀𝒆𝒏 𝟒(𝟑𝒙 − 𝒚𝟐) 510 𝟏𝟐𝒙 − 𝟒𝒚𝟐 (256 + 128 + 64 + 32 + 16 + 8 + 4 + 2) 8 -24 𝒊=𝟏 𝟖 𝒙 𝒊

One last example. Example For 𝑥 + 𝑦 2 , simplify and substitute for 𝑥 = 1 and 𝑦 = 2. Let’s try with substitution first. (1 + 2)2 = 9 How about expanding first? 𝑥2 + 𝑦2? Mmm… 12 + 22 = 5 Well that didn’t work. 𝑥 + 𝑦 2 ≠ 𝑥2 + 𝑦2. So what is it?

𝑥2 𝑥𝑦 𝑦𝑥 𝑦2 Let’s look at (x + y) 2 as an area. 𝑥 𝑦 𝑥 Here we want the total area. I made the size of 𝑥 look different to 𝑦 but this doesn’t matter. 𝑦𝑥 𝑦 𝑦2

So if we add each part together we get… 𝑥 + 𝑦 2 = 𝑥2 + 𝑥𝑦 +𝑦𝑥 + 𝑦2 = 𝑥2 + 2𝑥𝑦 + 𝑦2 How about 𝑥 – 𝑦 2 ? And how about (𝑥 + 𝑦)(𝑥 – 𝑦)?