Slideshow 9, Mathematics Mr Richard Sasaki Collecting Like Terms Slideshow 9, Mathematics Mr Richard Sasaki

Objectives Recall Algebraic Operations and notation To gather numbers and unknowns To simplify expressions with various unknowns

Algebraic Operations Simplify the following… 2×𝑥 2𝑥 = 𝑥+𝑥 2𝑥 = 𝑥×𝑦 𝑥𝑦 𝑥 2 + 𝑥 2 2 𝑥 2 = 𝑥×𝑥 𝑥 2 = 𝑥+𝑦 𝑥+𝑦 = 5𝑥 3 or 5 3 𝑥 Note: 5÷3 ×𝑥= . We shouldn’t write… . 1 2𝑥 3 or 1 2 3 𝑥 Decimal numbers like 1.2𝑥 are fine!

Vocabulary Let’s look at some components of the expression below. Why 6𝑥? 4𝑥+2𝑥 6𝑥 = We did 2+4=6. Operator (Plus) Coefficients Coefficients - Numbers that appear in front of unknowns. Operator - Symbols like +,−,×,÷, , that allow us to perform calculations. Symbols like % or # are not operators (in mathematics).

Collecting Like Terms 4𝑥+ 𝑥 = 5𝑥 1 Both 4𝑥 and 2𝑥 have the like term 𝑥. We can add the 𝑥 terms together (or subtract) as they are the same thing. So 4𝑥+2𝑥=6𝑥. Calculate 2 bananas + 4 bananas. 6 bananas It doesn’t matter if we add unknowns, bananas, objects, or numbers, we can collect them together. 4𝑥+ 𝑥 = 5𝑥 1 Here, we did the calculation 4+1=5. Where did the 1 come from? We don’t usually write the coefficient 1 though.

Collecting Like Terms 4𝑥+3𝑦 4𝑥+3𝑦 = Let’s try some terms that are not like terms. 4𝑥+3𝑦 4𝑥+3𝑦 = Can we add bananas to coconuts to make some magical banana coconuts? No! We can’t add 𝑥 to 𝑦 either. We can only combine like terms. So we just leave it the same. Note: If a question says simplify 𝑥+𝑦, write 𝑥+𝑦. If you write nothing you’ll get it wrong.

Collecting Like Terms + 5𝑥+2𝑦+6𝑥= 5𝑥+6𝑥+2𝑦 =11𝑥+2𝑦 2𝑥+3𝑦−𝑦+4𝑥= Examples Note: We can rearrange. 𝑎+𝑏=𝑏+𝑎. Simplify 5𝑥+2𝑦+6𝑥. 5𝑥+2𝑦+6𝑥= 5𝑥+6𝑥+2𝑦 =11𝑥+2𝑦 Simplify 2𝑥+3𝑦−𝑦+4𝑥. + 2𝑥+3𝑦−𝑦+4𝑥= 2𝑥+4𝑥+3𝑦−𝑦 =6𝑥+2𝑦 Note: The first term is positive unless there is a minus symbol.

Collecting Like Terms −7𝑥+9𝑦−5𝑦+2𝑧= −7𝑥+4𝑦+2𝑧 Example Simplify −7𝑥+9𝑦−5𝑦+2𝑧. −7𝑥+9𝑦−5𝑦+2𝑧= −7𝑥+4𝑦+2𝑧 It is usually better to write terms in alphabetical order. Notice the first term above is 𝑥, then 𝑦 and lastly, 𝑧. There is one exception however for binomials (an expression with two terms)… It is preferable to write −𝑥+𝑦 as . 𝑦−𝑥 To some people, it looks untidy to start with −.

8𝑦 2𝑦 7𝑥 𝑦 4𝑥 11𝑦 10𝑥 6𝑥 8𝑥+3𝑦 3𝑥+8𝑦 7𝑥−4 2𝑦+10 3𝑥+2𝑦 6𝑥+5𝑦 11𝑥+2𝑦 8𝑥+3𝑦 2𝑧−2𝑦 14𝑦 −4𝑥 10𝑥+5𝑦−7𝑧 2𝑎+3𝑏−3 3𝑥−3𝑦 3𝑦−𝑥 −𝑥−3 3.8𝑦−2𝑥 2𝑦−101𝑥 4𝑥 𝑥 29𝑥 35 3𝑥 4

Other Forms As you know, 𝑥 is an unknown. In the same way, 𝑥 2 is an unknown, especially if we don’t know 𝑥. For this reason, we can add and subtract like terms for 𝑥 2 or 𝑥 3 together (but not with each other). Note: 𝑥 3 is read 𝑥 cubed. Example Simplify 𝑥 3 + 𝑥 2 +2 𝑥 3 . 𝑥 3 + 𝑥 2 +2 𝑥 3 = 3 𝑥 3 + 𝑥 2 Note: Terms should be ordered …+ 𝑥 3 + 𝑥 2 +𝑥+𝑛.

2 𝑥 3 𝑥 3 +2 𝑥 2 3 𝑥 2 −𝑥 5 𝑦 2 −4𝑦 5 𝑥 2 +2𝑥−3 7 𝑥 6 +2 𝑥 4 2𝑥−4 𝑥 2 3 𝑥 2 − 𝑥 3 𝑥 2 +3 3𝑎𝑏 𝑎𝑏 𝑥𝑦 𝑎 2 𝑏 3𝑎 𝑏 2 +2𝑎𝑏 8𝑥 𝑐 𝑚 2 4𝑥+9𝑦 4𝑥+5𝑦+4𝑧

2 𝑎 2 𝑏−𝑎 𝑏 2 3𝑎𝑏−3𝑎+2 −3 𝑎 2 −4𝑎+10 2𝑎𝑏 6 −4 𝑎 2 −2𝑎 6 𝑥 2 −𝑥 4𝑎𝑏+𝑎 𝑏 2 34 𝑥 2 −42 6 𝑎 7 −6 𝑎 −7 2 𝑥 2 −5 4𝑥 𝑦 2 +13𝑥𝑦 3 𝑥 2 +7 𝑥 2 +7𝑥 8𝑥𝑦−6𝑥+𝑦 9 𝑎 2 −2𝑎 12 𝑥 2 𝑐 𝑚 2 24𝑥 𝑐𝑚