Contents:
Simple Rules: Operation signs, used when multiplying or dividing; or when adding or subtracting and two signs are together.
+ + makes + ie. 4 x 2 = 8 and. 4 + 2 = 6.
+ - makes - ie. 4 x -2 = -8 and. 4 + -2 = 2.
- + makes - ie. -42 = -2 and. 4 - +2 = 2.
- - makes + ie. -4When letters are multiplied together it is normal the leave out the x operator; m x n would be seen as mn.
Algebraic Expressions, using Letters: An algebraic expression contains letters as well as numbers. An expression is the mathematical operations necessary to solve the problem (a formula). The letters are present just to substitute for a number, the expression is true for a range of numbers therefore the letters are variable.
Example: Going shopping for 3 loaves of bread, 6 eggs and 2 pints of milk. If bread costs 50, eggs cost 20, and milk is 30 we could make an algebraic expression: b = bread, e = eggs, m = milk:
Shopping costs = 3 x b + 6 x e + 2 x m. or 3 x 50 + 6 x 20 + 2 x 30.
In algebra the times 'x' may be left out, leaving: 3b + 6e + 2m.
Example: Child A has 5 teddies more than child B. We don't know how many teddies they have, but we could write an expression for when it becomes known: if child B has 't' amount of teddies then child A would have 5 + 't' amount of teddies.
The total amount could be expressed as: 2t + 5 (both have 't' amount and the 5 extra).
Example: Children buy sweets at 'x' pence each, 10 sweets are priced 5x.
Example: Child A thinks of a number, multiplies it by 8, subtracts 12 then divides by 8. Child B forms a formula that will give the answer when the original number is known.
Original number: n
Multiplied by 8: 8n
Subtract 12: (8n - 12)
Divide by original number: (8n - 12)The expression (formula) is (8n -12)
Building up an Algebraic Expression: This is necessary when a question is set before a calculation can be made.
Example: A pile of 'b' number of bricks weigh 500Kg, how much would a pile of 'c' number of bricks weigh.
First consider what 1 brick weighs: 500
Example: The three angles of a triangle are x
+ y
The angles total to 180
x = 180 - y + z This is also known as an algebraic equation.
Example: A car travells at 'x' kpm for 2 hours then at 'y' kpm for 3 hours, how far has it travelled.
x for 2 hours and y for 3 hours = 2x + 3y kilometers.
Example: A school exam is attended by 'g' girls and 'b' boys, if the girls average mark is 'x' and the boys average mark is 'y', what is the total average.
Girls total = gx, boys total = by. Average is total mark
Substitution: The letters in an algebraic expression are substituted for numbers and the calculation can be completed. The numbers are constants, and called the co-efficients, the letters are called variables because under different circumstances they may have different values.
Rule for equations: any operation done on one side must also be done on the other side.
Example: Cost = 3b + 6e + 2m (from the bread, eggs and milk). b = 50, e = 20 and m = 30.
By substituting numbers for the letters give: 3 x 50 + 6 x 20 + 2 x 30, and a total cost calculated of: 330.
Example: t = (8n - 12)
New equation: t = (8 x 6 - 12)
Example:
=
+
Find R when S = 8 and T = 4.
Substitute the numbers for the letters:
+
. Add the fractions:
.
Invert both sides of the equation, therefore R = 2
.
Expansion from Brackets: The order of operations are powers then mutiplying or dividing then adding or subtracting. The change this order brackets are inserted. The operations inside the brackets must then be done first. The process of removing the brackets is called expansion.
Example: Expand and simplify: 9(2x - y). The 2x and the y are terms, the 9 is a factor. A factor operates on both terms; hence the expanded expression is: 18x - 9y.
Example: Expand and simplify: 9(4x + 4y) + 3(2x + 2y).
Answer: Each term is multiplied by its factor: (36x + 36y) + (6x + 6y).
Remove brackets and add like terms: 42x + 42y.
Example: Expand and simplify: 9(4x + 4y) - 3(2x - 2y).
Answer: Each term is multiplied by its factor: (36x + 36y) - (6x - 6y).
Remove brackets: care must be taken with a minus outside the brackets as it changes the sign of the terms within the brackets when the brackets are removed: 36x + 36y - 6x + 6y. Add the like terms: 30x +42y.
Example: Expand and simplify: (y +7)(y - 4).
Answer: Each term is within brackets and so is a factor of the other. There is no sign given therefore always assume it is a multiplication.
Holding the first bracket; multiply out: (y + 7)y - (y + 7)4. The first bracket is multiplied by the 'y' in the second bracket, and then by the -4 in the second bracket. Next the terms within the two brackets are multiplied by their own factor: (y
+7y) - (4y + 28).
Removing the brackets give: y
Another way of dealing with this is the 'smile rule' shown below:
= y
Example: Expand: (a - b)(a + b).
Answer: a
Exercise: Note: Enter a * for a
Factorisation into Brackets: is the opposite operation to expansion. The product of two or more simpler expressions may be factorised.
Example: Factorise: (4x
Answer: There are two terms, find what is common to both: (multiples of 4 and x). Divide each term by 4 and x and take them outside the brackets leaving: 4x(x + 2y).
4x multiplied by the first term, x = 4x
Example: Factorise: (9x
Answer: These two terms are both squared, the square root of each term may then be taken outside the brackets: 3x + y leaving (3x - y) within the brackets leaving: (3x + y)(3x - y). Expanding checks with the original problem.
Example: Factorise: y
Answer: Find two factors of 12 that sum to 7; 4 and 3. These go into separate brackets with each of the y from y
Example: Factorise: 6y
Answer: Similar to the above example but factorise the 6, into 3 and 2. The factors of -12 which sum to 1, are -4 and 3; this leaves: (3y - 4)(2y + 3). Check back to: 6y
Always check by expansion, the correct factors in the wrong brackets give the wrong answer: (3y + 3)(2y - 4) expands to: 6y
Example: Factorise: 3yz - 12y - 4x + xz.
Answer: Group common factors together: factor out z : 3yz + xz = (3y + x)z. Then factor out the numbers: -12y - 4x = (3y + x)-4.
Leaving: -4(3y + x) z(3y + x). Divide out the common factor (3y + x) leaves: (z - 4)(3y + x). Always check by expansion: 3yz + xz -12y -4x.
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Top of Factorisation
Exercise: Factorisation of quadratic expressions:
Note: Enter brackets, numbers, letters and signs without a space. Example answer: (x-6)(x+6)
Changing the Subject of a Formula: the subject letter in a formula is one that is expressed as equalling a series of numbers and other letters. If this subject letter is changed to another in the formula the formula need to be changed to show this. Rules covering rearranging are covered in Equations.
Example: If 'x' = 2y what would 'y' equal. To rearrange a formula or expression where one side is equal to the other side the same opperations have to be applied to both sides.
Therefore: x = 2y, if both sides are divided by 2 it would leave y the subject of the formula: leaving: x
Example: x = (y + z)
Answer: Clear the
Then subtract the z from both sides giving: 2x - z = y, or y = 2x - z.Example: r =
3V
h, make V the subject of the formula.
Answer: First clear the square root by squaring both sides: r
The multiply both sides by
Finally divide both sides by 3 to clear the V: V = (Example: ax + c = bx + d, make x the subject of the formula.
Answer: Get both the x's together onto one side, subtract bx and c from both sides:
ax - bx = d - c.
Next factorise the ax - bx: equals: x(a - b) = d - c.
Finally divide both sides by (a - b): leaving: x = (d - c)Note: It is important to show brackets in these examples where operations are mixed to show that some additions or subtractions have to be done before a multiplication or divide. Without brackets the answers could be wrong.
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