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ALGEBRAExpressions and
Equations
- Linear number patterns are sequences of numbers where the difference between terms is always the same (constant) - Rule generating a linear pattern is: Difference × n ± a constant
e.g. Write a rule (using n) to describe the following number patterns.
n Number of Squares (s)
Number of Dots
(d)
1 1 5
2 4 9
3 7 13
4 10 17
5 13 21
+ 3+ 3+ 3+ 3
+ 4
+ 4
+ 4
+ 4
Rule: s =
Rule: d =3×n 4×n
3×1= 33 = 1- 2
- 2
4×1= 44 = 5+ 1
+ 11. Find the difference between terms and if the same multiply by n
2. Substitute to find constant3. Check if rule works
3×4 – 2
4×4 + 1
FINDING RULES FOR LINEAR PATTERNS
CREATING EXPRESSIONS
Note: × and ÷ are not often used in Algebra
i.e. 5 × x = 5x i.e. 8 ÷ x = 8
x
Also a dot ‘.’ means multiply i.e. 2x . 2y = 2x × 2y
- Using suitable symbols to express rules
e.g. Write an expression for each of the following
a) A number with 12 added to it
b) A number with 9 subtracted from it
c) A number multiplied by 2
d) A number divided by 6
As long as you explain what a symbol represents, any symbol can be used
Let n = a number
n + 12
n - 9
n × 2 Best written as 2n
n ÷ 6 Best written as n 6
e.g. John has x dollars. How much will he have if:
a) He spends $35
b) He is given $28
c) He doubles his money
d) He spends half
x - 35
x + 28
x 2
x2
Once you have an expression, it can be used to calculate values if you know what the ‘variable’ (symbol) is worth.
e.g. John has $50, use the expressions to calculate
how much he will have in each situation:
50
- 35
50
+ 28
x 50 2
50 2
= $15
= $78
= $100
= $25
- ALL terms can be multipliedRules: 1) Multiply all numbers in the expression
2) Place letters in alphabetical order behind producte.g. Simplify:
a) 4c × 2d
b) –p × -2q × -6r
No number = 1 i.e. -p = -1p
= 4 × 2 × c × d= 8cd
= -1 × -2 × -6 × p × q × r= -12pqr
SIMPLIFYING EXPRESSIONS BY MULTIPLYING
POWERS- Remember: 3 × 3 × 3 ×
3 = 34
- Variables (letters) that are multiplied by themselves are treated the same way
e.g. Simplify these expressions that are written in full
a) r × r b) p × p × p × p × p
= r2 = p5
- Sometimes there may be two or more variables
e.g. Simplify
a) a × a × b × b × b
b) d × e × e × d × f
= a2b3 = d2e2fLetters should still be written in alphabetical order!
FOUR POWER RULES1. Multiplication
- Does x2 × x3 = x × x × x × x × x ?
YES- Therefore x2 × x3 = x5
- How do you get 2 3 = 5 ?+
- When multiplying index (power) expressions with the same letter, ADD the powers.
e.g. Simplify
a) p10 × p2 b) a3 × a2 × a= p(10 + 2) = a(3 + 2 + 1)
= p12 1
No number = 1 i.e. p = 1p1
= a6
- Remember to multiply any numbers in front of the variables first
e.g. Simplify
a) 2x3 × 3x4 b) 2a2 × 3a × 5a4
= 2 × 3 = 2 × 3 ×
5= 6
1
= 30
x(3 + 4)
a(2 + 1 + 4)x7
a7
2. Division
YES- Does 6 = 1 ? 6
- Therefore x = 1 x- Does x5 = x × x × x × x
× x ? x3 x × x × x
YES = x × x × 1 × 1 × 1
- Therefore x5 = x2
x3
- When dividing index (power) expressions with the same letter, SUBTRACT the powers.
e.g. Simplify
a) p5 ÷ p = p(5 - 1) = x(7 - 4)
= p4
= x3
- Remember to divide any numbers in front of the variables first
e.g. Simplify
a) 12x5 ÷ 6x4
= 12 ÷ 6= 2
- How do you get 5 3 = 2 ?-
1 b) x7
x4
If the power remaining is 1, it can be left out of the answer
x(5 - 4) b) 5a7
15a2
÷ 5÷ 5
= 1 5
a(7 - 2)
= 1 5
xa5 or a5
5
3. Powers of powers
- Does (x2)3 = x2 × x2 × x2 ? YES
- Therefore (x2)3 = x6
- How do you get 2 3 = 6 ?×
- Does x2 × x2 × x2 = x6 ? YES
- When taking a power of an index expression to a power, MULTIPLY the powers
e.g. Simplify
a) (c4)6 b) (a3)3
= c(4 × 6) = a(3 × 3)
= c24 = a9
- If there is a number in front, it must be raised to the power, not multipliede.g. Simplify
a) (3d2)3 b) (2a3)4× d(2 × 3)
= 27 a12= 33 × a(3 × 4)= 24
d6 = 16- If there is more than one term in the brackets, raise all to the power
e.g. Simplify
a) (x3y z4)3
= x(3 × 3)
= x9
1 y(1 × 3)z(4 × 3)
y3z12
b) (4b2c5)2 b(2 × 2)
= 16
c(5 × 2)
b4c10
= 42
4. Powers of zero- Any base to the power of zero has a value of 1
e.g. x0 = 1SQUARE ROOTS
- Simply halve the power (as √x is the same as x½ )
e.g. Simplify
a) = 10x4
b)= 8x3
LIKE AND UNLIKE TERMS- LIKE terms are those with exactly the same letter, or
combination of letters and powers
LIKE terms: UNLIKE terms:
2x, 3x, 31x4ab, 7ab
2x, 35x, 6x2
2ab, 2ace.g. Circle the LIKE terms in the following groups:
a) 3a 5b 6a 2c b) 2xy 4x 12xy 3z 4yx
While letters should be in order, terms are still LIKE if they are not.
100x8
64x6= 100 x
8 = 64 x6
SIMPLIFYING BY ADDING/SUBTRACTING- We ALWAYS aim to simplify expressions from expanded to
compact form- Only LIKE terms can be added or subtracted- When adding/subtracting just deal with the numbers in front of the
letterse.g. Simplify these expanded expressions into compact form:
a) a + a + a
b) 5x + 6x + 2x
c) 3p + 7q + 2p + 5q
= (1 + 1 + 1)a= 3a
= (5 + 6 + 2)x= 13x
= (3 + 2)p= 5p + 12q
d) 4a + 3a2 + 7a + a2
1 1 1
(+ 7 + 5)q
1 = (4 + 7)a
(+ 3 + 1)a2= 11a + 4a2
- For expressions involving both addition and subtraction take note of signs
a) 4x + 2y – 3x
b) 3a – 4b – 6a + 9b
c) 3x2 - 9x + 6x2 + 8x - 5
= (4 – 3)x= x + 2y = (3 -
6)a= -3a + 5b
= 9x2 - x - 5
d) 4ab2 +2a2b – 5ab2 + 3ab
(- 9 + 8)x
= -ab2 + 2a2b + 3ab
+ 2y(- 4 + 9)b
= (3 + 6)x2 - 5
If the number left in front of a letter is 1, it can be left out
e.g. Simplify the following expressions:
EXPANDING EXPRESSIONS- Does 6 × (3 + 5) = 6 × 3 + 6
× 5 ? 6 × 8 = 18 + 30 48 = 48
YES
- The removal of the brackets is known as the distributive law and can also be applied to algebraic expressions
- When expanding, simply multiply each term inside the bracket by the term directly in front
e.g. Expand
a) 6(x + y) b) -4(x – y)
c) -4(x – 6) d) 7(3x – 2)
e) x(2x + 3y) f) -3x(2x – 5)
= 6 × x
+ 6 × y =
6x
= -4 × x
- -4 × y = -4x
= -4 × x - -4 × 6 = -4x
= 7 × 3x
- 7 × 2 = 21x
= x × 2x
+ x × 3y =
2x2
1 1
= -3x × 2x
- -3x × 5 = -
6x2 Don’t forget to watch for sign changes!
+ 6y + 4y
+ 24 - 14
+ 3xy + 15x
- If there is more than one set of brackets, expand them all then collect any like terms.
e.g. Expand and simplify
a) 2(4x + y) + 8(3x – 2y)
b) -3(2a – 3b) – 4(5a + b)
= 2 × 4x
+ 2 × y
+ 8 × 3x
- 8 × 2y =
8x + 2y +
24x - 16y
= 32x - 14y
= -3 × 2a - -3 × 3b
- 4 × 5a
+ -4 × 1b = -
6a + 9b - 20a - 4b
= -26a
+ 5b
SUBSTITUTION- Involves replacing variables with numbers and calculating the
answer- Remember the BEDMAS rules
e.g. If m = 5, calculate m2 – 4m - 3
= 52 – 4×5 - 3= 25 – 4×5 - 3= 25 – 20 - 3= 2
- Formulas can also have more than one variable
e.g. If x = 4 and y = 6, calculate 3x – 2y
= 3×4 - 2×6= 12 - 12= 0
e.g. If a = 2, and b = 5, calculate 2b – a 4
Because the top needs to be calculated first, brackets are implied
= (2 × 5 – 2) 4= (10 – 2) 4= 8 4
= 2
ALGEBRAIC FRACTIONS
- Are fractions with letters in them. - Should be treated exactly like normal fractions.
1. Simplifying- As with normal fractions, look for common numbers and letter to
cancel out. Do numbers first, then letters
e.g. Simplify:
a) 3a 6a
÷ 3÷ 3
= 1 2
1
1
a(1 – 1)
= 1 2
b) 18x 6y
÷ 6÷ 6
= 3 1
xy
= 3x y
c) 4x3
6x2
÷ 2÷ 2
= 2 3
x(3 – 2)
= 2 3
x
2. Multiplying Fractions- Multiply top and bottom terms separately then simplify.
a) b × b2
2 5
e.g. Simplify:
= b3
10
b) y2 × 4 3 y
= 4y2
3y= 4y 33. Dividing Fractions
- Multiply the first fraction by the reciprocal of the second, then simplify
Note: b is the reciprocal of 2 2 b
a) 2a ÷ a2
5 3
e.g. Simplify:
= 6a 5a2
= 6 5a
or 6a-1
5
= b × b2
2 × 5= y2 × 4
3 × y
= 2a 5
× 3a2
= 2a × 35 × a2
4. Adding/Subtracting Fractions a) With the same denominator:
- Add/subtract the numerators and leave the denominator unchanged. Simplify if possible.
a) 3x + 3x
10 10
e.g. Simplify:= 3x +
3x 10= 6x 10÷ 2÷ 2
= 3x 5
b) 6a - b 5 5
= 6a - b 5
b) With different denominators: - Multiply denominators to find a common
term. - Cross multiply to find equivalent numerators. - Add/subtract fractions then simplify.
e.g. Simplify:a) a +
2a 2 3= 3a + 4a
6= 7a 6
b) 2x – 5x
3 4= 8x – 15x 12= -7x 12
= 2×3
3×a + 2×2a = 3×4
4×2x - 3×5x
FACTORISING EXPRESSIONS- Factorising is the reverse of expanding- To factorise: 1) Look for a common factor to put outside the
brackets2) Inside brackets place numbers/letters needed to make up original terms
e.g. Factorise
a) 2x + 2y b) 2a + 4b – 6c
= 2( ) x + y = 2( )
a + 2b
e.g. Factorise
a) 6x - 15 b) 30a + 20
= 3( ) 2x
- 5 = 10( ) 3a + 2
- Always look for the highest common factor
You should always check your answer by expanding it
e.g. Factorise
a) 6x + 3 b) 20b - 10= 3( ) 2x
+ 1 = 10( ) 2b - 1
- Sometimes a ‘1’ will need to be left in the brackets
- 3c
e.g. Factorise
a) cd - ce b) xyz + 2xy – 3yz = c( ) d - e = y( )
xz + 2x
- Letters can also be common factors
e.g. Factorise
a) 5a2 – 7a5 b) 4b2 + 6b3= a2( ) 5 - 7a3 = 2b2( ) 2 + 3b
- Powers greater than 1 can also be common factors
- 3z
c) 4ad – 8a
= 4 ( ) a
d
- 2
SOLVING EQUATIONS- When solving we need to isolate the unknown variable to find its
value- To isolate we work backwards by undoing operations
1) To undo multiplication we use divisione.g. Solve 3x = 18
÷3 ÷3x = 6
2) To undo addition we use subtractione.g. Solve x + 2 = 6
-2 -2x = 4
3) To undo subtraction we use additione.g. Solve x - 8 = 11
+8 +8x = 194) To undo division we use
multiplication
×5x = 30
e.g. Solve x = 6 5
×5
- Terms containing the variable (x) should be placed on one side (often left)e.g. Solve
a) 5x = 3x + 6
b) -6x = -2x + 12-3x -3x
2x = 6÷2÷2
x = 3
You should always check your answer by substituting into original equation
- Does 5×3 = 3×3 + 6 ?
15 = 9 + 6
YES
+2x
+2x -4x = 12
÷-4
÷-4 x = -3
Always line up equals signs and each line should contain the variable and one equals sign
- Does -6×-3 = -2×-3 + 12 ?
18 = 6 + 12
YES
- Numbers should be placed on the side opposite to the variables (often right)e.g. Solve
a) 6x – 5 = 13
b) -3x + 10 = 31+5 +5
6x = 18÷6÷6
x = 3
-10 -10 -3x = 21
÷-3
÷-3 x = -7
Always look at the sign in front of the term/number to decide operation
Don’t forget the integer
rules!
- Same rules apply for combined equations
e.g. Solvea) 5x + 8 = 2x + 20
b) 4x - 12 = -2x + 24-2x -2x
3x + 8 = 20 -8-8
3x = 12
+2x
+2x 6x - 12 =
24
÷6÷6 x = 6
÷3÷3 x = 4
+12
+12 6x = 36
- Answers can also be negatives and/or fractions
e.g. Solvea) 8x + 3 = -12x - 17
b) 5x + 2 = 3x + 1+12
x+12x 20x + 3 = -
17 -3-3 20x = -20
-3x -3x 2x + 2 = 1
÷2÷2 x = -1 2
÷20
÷20 x = -1
-2-2 2x = -1
Make sure you don’t forget to leave the sign
too!
Answer can be written as a decimal but easiest to leave as a
fraction
- Expand any brackets first
e.g. Solve
a) 3(x + 1) = 6 b) 2(3x – 1) = x + 8
-3-3 3x = 3
÷3÷3 x = 1
3x+ 3= 6 6x- 2 = x + 8 -x-x
5x - 2 = 8 +2+2
5x = 10÷5÷5
x = 2- For fractions, cross multiply, then solve
e.g. Solve
a) x = 9
4 2
2x= 36÷2÷2
x = 18
b) 3x - 1 = x + 3
5 22(3x - 1)= 5(x +
3)6x
- 2 = 5x+ 15-5x-5x
x - 2 = 15 +2+2
x = 17
- For two or more fractions, find a common denominator, multiply it by each term, then solve
e.g. Solve 4x - 2x = 10
5 3
5 × 3 = 15
×15
×15
×15
60x
5
- 30x
3
= 150
Simplify terms by dividing numerator by denominator12
x
- 10x
= 150
2x = 150 ÷2÷2
x = 75
WRITING EQUATIONS AND SOLVING- Involves writing an equation and then solvinge.g. Write an equation for the following information
a) I think of a number, multiply it by 3 and then add 12. The result is 36.
a) I think of a number, multiply it by 5 and then subtract 4. The result is
n3 + 12
= 36
Let n = a number
Let n = a number
n5 - 4 = n + 18
the same as if 18 were added to the number
e.g. Write an equation for the following information and solvea) A rectangular pool has a length 5m longer than its width. The
perimeter of the pool is 58m. Find its width
Draw a diagram
Let x = width
-10-10
Therefore width is 12 m
x x
x + 5
x + 5
x + 5 + x + x + 5 + x = 58 4x + 10 =
58 4x = 48
÷4÷4x = 12
b) I think of a number and multiply it by 7. The result is the same as if I multiply this number by 4 and add 15. What is this number?
Let n = a number
n = n
+ 15-4n-4n
3n = 15
Therefore the number is 5
7 4
÷3÷3n = 5
- Inequations contain one of four inequality signs: < > ≤ ≥ - To solve follow the same rules as when solving equations- Except: Reverse the direction of the sign when dividing by a negative
e.g. Solvea) 3x + 8 > 24
b) -2x - 5 ≤ 13-8-8
3x > 16 ÷3÷3x > 16 3
+5+5-2x ≤ 18
÷-2
÷-2
Sign reverses as dividing by a negative
x ≥ -9
As answer not a whole number, leave as a fraction
SOLVING INEQUATIONS
CHANGING THE SUBJECT- Involves rearranging the formula in order to isolate the new ‘subject’- Same rules as for solving are
usede.g.
a) Make x the subject of y = 6x - 2 +2+2
y + 2 = 6x÷6÷6
y + 2 = x 6
All terms on the left must be divided by 6
b) Make R the subject of IR = V÷I÷I
Treat letters the same as numbers!R = V
Ic) Make x the subject of y = 2x2
÷2÷2y = x2
2
Taking the square root undoes squaringy
2=x
Remember: When rearranging or changing the subject you are NOT finding a numerical answer