The greatest mathematical tool of all!! An introduction to Algebra The greatest mathematical tool of all!!

This is a course in basic introductory algebra. Essential Prerequisites: Ability to work with directed numbers (positives and negatives) An understanding of order of operations

Introduction Stephen is 5 years older than Nancy. Their ages add to 80. How old are they? Without algebra, students would probably guess different pairs of values (with each pair differing by 5) and hope to somehow find a pair that add to 80. Eventually, we might work out that Stephen is 42 ½ and Nancy is 37 ½ . The problem, however, is that there can be too much guesswork. Algebra takes away the guesswork. SEE THE SOLUTION!

Contents 1. Substituting – numerals and pronumerals GO!! 2. Like and unlike terms – adding and subtracting GO!! 3. Multiplying GO!! 4. Dividing GO!! 5. Mixed operations and order of operations GO!!

Section 1 Substituting – numerals and pronumerals

When working with algebra, you will meet TWO different kinds of terms….. NUMERALS These are all the ordinary numbers you’ve been working with all your life. Numerals include 2, 5, 7, 235, 15½, 9¾, 2.757, 3.07, – 9, – 7.6 , 0,  and so on. PRONUMERALS These are symbols like , , , and letters (either single letters or combinations) like x, y, a, b, ab, xyz, x2 , y3 etc…Pronumerals often take the place of numerals.

example 1 = 7 (ans) = – 4 (ans) If a = 5 b = 2 c = 3 find the value of (2) c – a – b SOLUTION SOLUTION a + b c – a – b = 5 + 2 = 3 – 5 – 2 = 7 (ans) = – 4 (ans)

THIS DOESN’T APPLY WHEN WORKING ONLY WITH NUMBERS NOTE WHEN WORKING WITH PRONUMERALS YOU’RE ALLOWED TO LEAVE OUT MULTIPLICATION SIGNS. PRONUMERALS ARE USUALLY WRITTEN ALPHABETICALLY (ab rather than ba) 3 x a = 3a 2 x p = 2p a x b = ab 5 x a x b = 5ab c x a x b = abc THIS DOESN’T APPLY WHEN WORKING ONLY WITH NUMBERS 3 x 4 can’t be written as 34!

= 13 (ans) = 24 (ans) If a = 5 b = 2 c = 3 find the value of (4) 4bc – 2a + ab SOLUTION SOLUTION ab + c 4bc – 2a + ab = 5 × 2 + 3 = 4 × 2 × 3 – 2 × 5 + 5 × 2 = 13 (ans) = 24 – 10 + 10 Remember to do multiplication first!! = 24 (ans)

= 25 (ans) = 8 (ans) If a = 5 b = 2 c = 3 find the value of (6) 4b(7c – 4a) SOLUTION SOLUTION a (b + c) 4b(7c – 4a) = a × (b + c) = 4 × b × (7 × c – 4 × a) = 5 × (2 + 3) = 4 × 2 × (7 × 3 – 4 × 5) = 5 × 5 = 8 × (21 – 20) = 25 (ans) = 8 × 1 = 8 (ans)

example 2 = 1 (ans) = 18 (ans) REMEMBER ORDER OF OPERATIONS If a = – 3 c = – 4 find the value of (1) 3a + b (2) c – (4a – b) SOLUTION SOLUTION c – (4a – b) = 3a + b = c – (4 × a – b) = 3 × a + b = 3 × – 3 + 10 = – 4 – (4 × – 3 – 10) = – 4 – ( – 12 – 10) = – 9 + 10 = – 4 – ( – 22) = 1 (ans) = – 4 + 22 = 18 (ans) Note a – (– b) is same as a + b !!

= 109 (ans) = 2.4 (ans) If a = – 3 b = 10 MULTIPLY c = – 4 BEFORE evaluate MULTIPLY BEFORE YOU ADD!!   (4) SOLUTION (3) a2 + b2   SOLUTION = a2 + b2   = a × a + b × b   = – 3 × – 3 + 10 × 10 = 9 + 100   = 109 (ans) = 2.4 (ans)

Section 2 Like and Unlike terms Adding and Subtracting

5×7 + 3×7 = 8×7 Work out the value of 5 × 7 + 3 × 7. 5 × 7 + 3 × 7 = 35 + 21 = 56 Now work out the value of 8 × 7. 8 × 7 = 56 So here we have two different questions that give the same answer, 56. So we can make this conclusion: 5×7 + 3×7 = 8×7

Can you predict the value of 9 × 5 – 2 × 5? 5 × 7 + 3× 7 = 8× 7 Or, in words, 5 lots of 7 + 3 lots of 7 = 8 lots of 7 Can you predict the value of 9 × 5 – 2 × 5? If you said 7 × 5 then you would be correct! Check that both sums equal 35!

“Long sum” “Short sum” Answer Try these! Make sure you write the SHORT SUM first, then the answer! “Long sum” “Short sum” Answer 2 x 8 + 5 x 8 7 x 8 56 6 x 9 + 2 x 9 4 x 7 + 1 x 7 8 x 9 – 3 x 9 7 x 6 – 3 x 6 5 x 2 + 2 8 x 9 72 5 x 7 35 5 x 9 45 4 x 6 24 12 6 x 2 Rewrite as 5 x 2 + 1 x 2

The general pattern 3 × 1 + 4 × 1 = 7 × 1 3 × 2 + 4 × 2 = 7 × 2 So by now you are hopefully beginning to see the general pattern. For example using the fact 3 + 4 = 7 we can write…. 3 × 1 + 4 × 1 = 7 × 1 3 × 2 + 4 × 2 = 7 × 2 3 × 8 + 4 × 8 = 7 × 8 3 × 9½ + 4 × 9½ = 7 × 9½ In fact, the pattern holds for all numbers (not just 1, 2, 8 and 9½) and can be written more generally as 3 × a + 4 × a = 7 × a or 3 × x + 4 × x = 7 × x or any pronumeral (letter) of your choice!

Now try these: 7 × b + 8 × b = 15 × b 2 × y + 9 × y = 11 × y 4 × p – 2 × p = 2 × p 6 × q – 1 × q = 5 × q 8 × x + 1 × x = 9 × x Remember x really means 1x 4 × x + x = 5 × x 8 × a – 2 × a = 6 × a

What would be a single statement that would cover all possibilities in this pattern? 8 × 7 + 5 × 7 = 13 × 7 8 × 2 + 5 × 2 = 13 × 2 Ans: 8 × a + 5 × a = 13 × a 8 × 12 + 5 × 12 = 13 × 12 And this pattern? Of course we could have used any pronumeral here – it does not have to be a or w. 5 × 9 – 2 × 9 = 3 × 9 5 × 2 – 2 × 2 = 3 × 2 5 × 79 – 2 × 79 = 3 × 79 Ans: 5 × w – 2 × w = 3 × w

And now for some good news When you’re writing algebra sums, you’re allowed to LEAVE OUT MULTIPLICATION SIGNS! So, 3 × c can be written as 3c 7 × a × b can be written as 7ab x × y can be written as xy 3 × 4 CAN’T be written as 34!! BUT

Now try these: 4a + 7a = 11a 5y + 8y = 13y 2p – 6p = – 4p q + 7q = 8q Remember x really means 1x x + x = 2x 10x + x = 11x a – 7a = – 6a – 2z – 7z + 5z =

When terms are multiplied, the order is not important…. 6 × 5 is the same as 5 × 6 (both = 30) a × b is the same as b × a. i.e. ab = ba. We usually use alphabetical order though, so ab rather than ba. a × b × c = b × a × c = a × c × b = c × a × b etc …. Again, we prefer alphabetical order so abc is best. 6 × a is the same as a × 6 i.e. 6a = a6 But the number is usually written first! Although it’s still correct, we don’t write a6. Always write 6a. So…. PUT THE NUMBERS BEFORE PRONUMERALS

More about like terms… We know that ab and ba are the same thing, so we can do sums like 4ab + 5ba = 9ab 7xy + 9yx = 16xy 3abc – 2bca + 8cab = 9abc mnp – 2mpn – 7pmn = – 8mnp Note that in each case, the number goes first alphabetical order is used for the answer (though it is still correct to write 9ba, 16yx etc…)

Like and Unlike Terms 6 × 3 + 5 × 7 ? Key Question: We know we can write 7 × 5 + 4 × 5 as a “short sum” 11 × 5, but is there a similar way of writing 6 × 3 + 5 × 7 ? If we calculate this sum, it is equal to 18 + 35 = 53 But there are no factors of 53 (other than 1 and 53) so there is NO SHORT SUM for 6 × 3 + 5 × 7 !! Because of this, we can conclude that there is no easy way of writing 6a + 7b, other than 6a + 7b! In summary, we can simplify 6a + 7a to get 13a. But cannot simplify 6a + 7b.

Like and Unlike Terms 6a + 7a is an example of LIKE TERMS. Like terms can be added or subtracted to get a simpler answer (13a in this case) 6a + 7b is an example of UNLIKE TERMS. Like terms cannot be added or subtracted.

Like and Unlike Terms You will encounter terms with powers such as x2 , 3a2 , 5p3, 3a2 b etc. These are treated the same way as terms with single pronumerals. x2 and x are UNLIKE, just as x and y are. 2a and 3a2 are UNLIKE and can’t be added or subtracted 3b and 3b4 are UNLIKE and can’t be added or subtracted 2ab and 4ab2 are UNLIKE and can’t be added or subtracted 2a2 and 3a2 are LIKE and can be added to get 5a2. Subtracted to get –1a2 or – a2 9a6 and 4a6 are LIKE and can be added to get 13a6. Subtracted to get 5a6 2a2band 4ba2 are LIKE and can be added to get 6a2b. Subtracted to get – 2a2b 2a2band 4ab2 are UNLIKE and can’t be added or subtracted. They’re unlike because the powers are on different pronumerals ab and ac are UNLIKE and can’t be added or subtracted 3a and 5 are UNLIKE and can’t be added or subtracted 3a and 3 are UNLIKE and can’t be added or subtracted

In the table below, match each term from Column 1 with its “like” term from Column 2 – 11pk2 6w 2kp2 6 7a ab 9bca 3y h 7pk2 2c2 – 7d 12c 5c 7d 3h – 5w 3p2k 5ba – 3c2 y 4abc 2 Answers next slide

In the table below, match each term from Column 1 with its “like” term from Column 2 – 11k2p 6w 2kp2 6 7a ab 9bca 3y h 7pk2 2c2 – 7d 12c 5c 7d 3h – 5w 3p2k 5ba – 3c2 y 4abc 2

Like terms – very important in addition and subtraction algebra sums ! 3a + 2a = 5a 6ab – 2ab = 4ab These questions have algebra parts that are like (the same). When that happens, you can simplify them! 7a2 – 3a2 = 4a2 2ac – 7ca = – 5ac 8xy2 – 3xy2 = 5xy2 x – 7x + 2x = – 4x 8x2 – 3x2 = 5x2 These questions have algebra parts that are different. When that happens, you can’t simplify them! 8x – 3y = 8x – 3y 5x2 – 3x = 5x2 – 3x 2ab – 3ac = 2ab – 3ac

A mixed bag. Which have like terms? Simplify those that do. 5a – 3a 2a 5a + 6 2x + 7x 9x x + 6x 7x 3x + 8y 8x + 8y 4a – 2b a – 7a – 6a 5a + 4 5abc + 4cba 9abc 2x + 7x2 2x3 + 2x2 3xy + 8yx 11xy xy + yx 2xy 4ab – 2b ab – ba 5a + 4a2 5 + 4a2 3x – 9x – 6x x – 9x – 8x –xy + 7yx 6xy 4x3 + 5x3 9x3 4ab – 7ba –3ab 7ac – ca 6ac

More advanced examples Simplifying expressions with more than two terms 3x + 5x + y + 8y The like terms are added together = 8x + 9y (ans) 2a – 3a + 5b – 6b The like terms are simplified = – a – b (ans) 5a2 + 3a + 2a2 + a The like terms are simplified = 7a2 + 4a (ans) Remember that terms with a and a2 are unlike and can’t be added

Rearranging terms Simplify 5 – 7 + 6 – 2 5 – 7 + 6 – 2 = – 2 + 6 – 2 using “cut ‘n’ paste” Simplify 5 – 7 + 6 – 2 5 – 7 + 6 – 2 Working left to right = – 2 + 6 – 2 = 4 – 2 = 2 But we can also rearrange the terms in the original question using “cut ‘n’ paste” First, draw lines to separate the terms, placing lines in front of each + or – sign 5 –7 +6 –2 Now, “cut” any term between the lines (with its sign) and move it to a new position. We’ll move the “+6” next to the “5”, swapping it with the “– 7” 5 –2

5 + 6 – 7 – 2 Now the original question appears as Which can now easily be simplified to the correct answer, 2. Note that we did not HAVE to cut and paste the +6 and the –7 . We are allowed to cut and paste ANY TERMS we like.

= 8a – 7b ans Example 1: Simplify 3a + 2b + 5a – 9b 3a +2b +5a –9b 3a We will now apply this to help us simplify ALGEBRAIC EXPRESSIONS, and aim to cut and paste so like terms are together. Example 1: Simplify 3a + 2b + 5a – 9b Here we’ll use cut ‘n’ paste to bring the a’s together and the b’s together by swapping the +2b and +5a 3a +2b +5a –9b 3a –9b 3a + 5a + 2b – 9b The question now becomes = 8a – 7b ans Simplifying like terms, we get 3a + 5a = 8a 2b – 9b = – 7b

= 9a – 11b ans Example 2: Simplify a – 9b – 2b + 8a a – 9b – 2b +8a a Again use cut ‘n’ paste to bring the a’s together and the b’s together by swapping the – 9b and +8a a – 9b – 2b +8a a –2b a + 8a – 2b – 9b The question now becomes = 9a – 11b ans Simplifying like terms, we get a + 8a = 9a – 2b – 9b = – 11b

= 6x + 3 ans Example 3: Simplify 2x – 5 + 4x + 8 2x – 5 +4x +8 2x +8 Again use cut ‘n’ paste to bring the x’s together and the numbers together by swapping the – 5 and +4x 2x – 5 +4x +8 2x +8 2x + 4x – 5 + 8 The question now becomes = 6x + 3 ans Simplifying like terms, we get 2x + 4x = 6x – 5 + 8 = +3

= 7y – x – 7x2 ans Example 4: Simplify 3y – 2x – 5x2 + 4y + x – 2x2 3y Here there are 6 terms which can be grouped into 3 pairs of like terms (2 terms contain an x, 2 terms contain an x2 and 2 terms contain y. 3y – 2x – 5x2 +4y +x – 2x2 Swap +4y with – 2x = 3y +4y – 5x2 – 2x +x – 2x2 This puts the y’s together Now swap +x with – 5x2 = 3y +4y +x – 2x – 5x2 – 2x2 This puts the x’s together and the x2 terms together Simplifying like terms, we get 3y + 4y = 7y x– 2x = – x – 5x2 – 2x2 = – 7 x2 = 7y – x – 7x2 ans

Section 3 Multiplying and working with brackets

Remember the basic rules Place numbers before letters Keep letters in alphabetical order Two negatives multiply to make a positive A negative and a positive multiply to make a negative If an even number of negatives is multiplied, the answer is a positive (because they pair off) If an odd number of negatives is multiplied, the answer is a negative (one is left after they pair off) You can rearrange terms that are all being multiplied (3 x 4 x 5 = 5 x 3 x 4 = 4 x 5 x 3; ab = ba etc…)

All algebraic terms can be multiplied When doing multiplication, we do not have to bother with like terms! 4 × a = 4a – x × –3y = 3xy a × 5 × 2 = 10a – 5x × –3y × – 6p = –90pxy c × a × b = abc w × – 3 × k = – 3kw x × a × b × w = abwx a × 3bc = 3abc – 2a × 3b × c × – 5d = – 3b × a = – 3ab 30abcd a × 5c = 5ac ½ a × 5b × 6c = 15abc 2a × b = 2ab NOTE in this last question, it’s easier to change the order and do ½a x 6c x 5b

When you multiply two or more of the same pronumeral….. a × a b × b × b – x × x × x = aa = bbb = – xxx = a2 = b3 = – x3 a2 × a b2 × b × b3 (– x)4 = aa × a = bbbbbb = (– x)(–x)(– x)(–x) = aaa = b6 = x4 = a3

Working with brackets….. = 3a × 3a 7b × 7b × 7b × 7b = 2401b4 = 9a2 (–5x)3 (– ab)4 = – 5x × – 5x × – 5x = –ab × –ab × –ab × –ab = – 125x3 = +a4b4 NOTE: negatives raised to an even power give a POSITIVE negatives raised to an odd power give a NEGATIVE

Mixed examples - multiplying QUESTION ANSWER 2a × b = –2a × –5a × a = 5a × 3b = –a × b × ab × 2ba = 2a × – 3 = –4a × 3a2 = a × – a = a2 × a3 = 2 × –a = –y × –y × –y = 3a × 2a = 2ab × –3a2b3 = –4p × –2p = –cd × –2cd = 2a × 4a × 7a = 2a × 3a × 5a = ab × 3ab = 2a + 3a + 5a = a × 3ab × 5b × 2 = (2ab)3 = ab × –2ab × –3a = (–5abc)2 = –a × – 3 × –2a × 6b (–2cdg)3 = 2ab 10a3 15ab –2a3b3 –6a –12a3 –a2 a5 –2a –y3 6a2 –6a3b4 8p2 2c2d2 56a3 30a3 3a2b2 11a 30a2b2 8a3b3 6a3b2 25a2b2c2 –36a2b –8c3d3g3 Note the blue one! It’s an addition!!

Section 4 Dividing

Consider the expression 24 ÷ 18. Basically, all expressions with a division sign can be simplified, or at least rewritten in a more concise form. Consider the expression 24 ÷ 18. This can be written as a fraction and simplified further by dividing (cancelling) numerator and denominator by 6…. 4 3

3 = 4x ans Example 1 Solution The same process can be applied to algebraic expressions…. 3 Example 1 Simplify 12x ÷ 3 Solution 12x ÷ 3 Writing as a fraction 4 Now think…. What is the largest number that divides into both numerator and denominator? (the HCF ) 1 Note when there is only a “1” left in the denominator, ignore it! = 4x ans

= 4b ans Example 2 Solution Simplify 8ab ÷ 2a 8ab ÷ 2a Writing as a fraction 4 Dividing numerator and denominator by HCF 2 and also by a. 1 = 4b ans

Example 3 Solution Simplify 28abc ÷ 18acd 28abc ÷ 18acd Writing as a fraction Dividing numerator and denominator by HCF (2) 14 and by a and by c. 9 Note – in this question (and many others) your answer will be a fraction!

Example 4 Solution Simplify 20a3b2c ÷ 8ab2c4 20a3b2c ÷ 8ab2c4 Writing as a fraction and in “expanded” format to make dividing easier Dividing numerator and denominator by HCF (4) 5 and cancelling matching pairs of pronumerals (a with a, b with b etc) 2

Example 5 Solution Simplify 5xy2z ÷ –15x2y3z5 5xy2z ÷ –15x2y3z5 Writing as a fraction and in “expanded” format to make dividing easier Dividing numerator and denominator by HCF (5) 1 and cancelling matching pairs of pronumerals (x with x, y with y etc) – 3 IMPORTANT NOTES When a negative sign remains in top or bottom, place it in front of the whole fraction When only a “1” remains in the top, you must keep it. (Remember when a “1” remains in the bottom, you can ignore it)

Section 5 Mixed Operations

When doing more complicated sums with a mixtured of the four operations (and brackets) you must observe the ORDER OF OPERATIONS RULES…. B Before anything, simplify all BRACKETS Then.. DM Working from left to right, do all DIVISON and MULTIPLICATION operations Then.. AS Working from left to right, do all ADDITION and SUBTRACTION operations

Example 1 Solution Simplify 20 – 2 × 9 20 – 2 × 9 = 20 – 18 = 2 (ans) No brackets. 20 – 2 × 9 Do the multiplication first = 20 – 18 = 2 (ans) Now do the subtraction NOTE – good setting out has all the “=“ signs directly under one another, and never more than one “=“ sign on the same line.

Example 2 Solution Simplify 13 – (6 + 5) × 4 13 – (6 + 5) × 4 Do brackets first. = 13 – 11 × 4 = 13 – 44 Now do the multiplication = – 31 (ans) Now do the subtraction

Example 3 Solution Simplify 2x × 3 + 4 × 5x 2x × 3 + 4 × 5x No brackets. 2x × 3 + 4 × 5x Do the two multiplications working left to right 2x × 3 + 4 × 5x = 6x + 20x = 26x (ans) Now do the addition Remembering that you can only add LIKE TERMS

Example 4 Solution Simplify 2x × 6xy – 4y × 3x2 2x × 6xy – 4y × 3x2 No brackets. 2x × 6xy – 4y × 3x2 Do the two multiplications working left to right 2x × 6xy – 4y × 3x2 = 12x2y – 12x2y = 0 (ans) Now do the subtraction

Example 5 Solution Simplify 12ab – (2b + 3b) × 4a Do brackets first. = 12ab – 5b × 4a = 12ab – 20ab Now do the multiplication = – 8ab (ans) Now do the subtraction

Example 6 Solution = 6a – 2a2 (ans) Simplify 8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2 Solution 8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2 Brackets first. = 8a – 12ab ÷ 6b + 3a × a – 5a2 Division & multiplication = 8a – 2a + 3a2 – 5a2 Now subtract like terms = 6a – 2a2 (ans)

Example 7 Solution Simplify (3ab2)2 – (ab + ab) × 4ab3 + 2a3b ÷ – ab Note (3ab2)2 = 3ab2 x 3ab2 = 9a2b4 (3ab2)2 – (ab + ab) × 4ab3 + 2a3b5 ÷ – ab Brackets first = 9a2b4 – 2ab × 4ab3 + 2a3b5 ÷ – ab = 9a2b4 – 8a2b4 – 2a2b4 Multiplication & Division Note 2ab x 4ab3 = 8a2b4 Note 2a3b5 ÷ – ab = – 2a2b4 Subtract as these are all like terms = –a2b4

The solution to our introductory problem on Slide #3 Let Nancy’s age = x So Nancy’s age is 37½ and Stephen (who is 5 years older) must be 42 ½ So Stephen’s age = x + 5 These add to 80, so x + 5 + x = 80 2x + 5 = 80 NO GUESSWORK!!! 2x = 75 x = 75 ÷ 2 BACK TO CONTENTS x = 37 ½