Tutorial On Algebra

A Math tutorial ---on Algebra This tutorial takes you through the basic operations of Algebra.You can use this to learn or to review. The first step is to learn the Algebraic Operations of addition,subtraction,multiplication and Division. Addition:

+3 +4 = 7 or +7 +3 + (-4 ) = 3 -4 =-1 [because 4 is a bigger number in the negative sense.] You are taking 3 and subtracting four...there is loss and a negative one.... Subtraction

3 - (+4 ) the rule is as follows: 3 -4 = -1 3 - (-4 ) = 7 If the signs inside and outside the bracket are the same,it means addition only;so 3- (-4 ) = 3 + 4 = 7 If the signs inside and outside the bracket are different,that is, one is positive and another negative,then take it as negative. Multiplication follows the same rule: simply 12

3 x 4 = (+3)x(+4 ) = 12 (-3) x (-4 ) = + 12 or (3) x (-4 ) = -12 (-3) x (4 ) = (-3) x

(+4 ) = -12 Note that if the sign ainside and outside the brackets are the same ,you get + (positive0 result...If they are different ,you get - or negative result. Division is similar:

12/3 = 4 -12/3=-4 -12/(-3) = 4 12/ (-3) = -4

Constants and variables: take the expression y = 2 X here '2' is a constant; x is a variable.....We can assign different numbers to X. When we say that Martin's age is twice that of his sister Susan. we write : y=2X Of course Y is the age of Martin while X is the age of Susan...we do not know now what are their respective ages... .Suppose Martin's age is given as 12...Then, we know that

Susan's age is 6 years.....We found X knowing Y....This process we call "solving an equation" -----Easy! Knowing Y,the quantity on the right side, we 'figure out' or solve the quantity or variable X in the left......The equality sign makes this statement an 'equation'. Take another equation as an example: Ihave $12 in my hand.If each box of chocolates cost $3, how many boxes of chocolate I can get for the money in hand: Set up the equation first: Number of boxes = N = y / x Y is th emoney on hand : y=12 X is the price of each box : x= 3 Therefore N= y/X = 12 / 3 = 4 Exponents Writing exponents for algebraic varaibles is a short -form. Suppose you want to multiply 2 five times; y = 2x2x2x2x2 = 32 5 You could write simply : y = 2 ^5 = 2 = 32 Now, 2 is called the base and 5 ,the exponent or power. We say that ' 2 is raised to the power of 5" Let us see some rules for using the exponential from of writing: 5

3

8

Suppose 2 x2 =2 The product is the same as multiplying 2 eight times over. Therefore the rule of multiplication [if the base is the same] is to add the exponents or powers. m n m+n Rule 1 X .X = X It is easy to write the reciprocals of numbers: -1 1/4 = 4 m n m Using this method: X / x = x . x What if m= n .Therefore

Then 0

-n =

x

m-n

the above expression becomes 1

x =1 Now we have an important and powerful expression in algebra....X can be nay number- If any number is raised to the power of zero, we get 1: Thus 0 0 0

10

= 22

= 24 000 = 1.

Combining the like terms Consider this simple example first. John has two fruit baskets.In one basket ,he has 8 oranges and 4 apples;in another basket,he has 3 oranges and 6 apples.What is the total number of oranges and apples? We shall denote orange by 'x' and apple by 'y'.The total fruits be called 'z' We shall write it out in algebraic terms: In basket 1, we have 8x and 4 y's. In Basket 2, we have 3x and 6y's. Then total Z = 8x + 4 y + 3x + 6y We need to collect the like terms--all the ornages and all the apples,that is x's and y's.We use brackets to collect the like terms. The algebraic expression is written as follows: Z = 8x + 3x + 4 y + 6y Z= (8 + 3) x + ( 4 + 6 )y Therefore Z = 11 x + 10 y This method we call 'combining like terms. You can work out the next problem yourself. z = 2.5 x + 3 y + 1.5 x - y - 2x + 0.5 y [The answer would be :

z = 2 x + 2.5 y ]

Expand the expression This is the reverse process we used in combining the like terms. Consider this example first: Linda bought books for 12 dollars and pencils for 8 dollars.She has to pay a sales tax at 8% for all the items. Work out the total tax. Sales Tax = ( 12 + 8 ) . (8/100) 1.6 $

= ( 20) . (8/100) = 20 x 8 /100 = 160/100 =

The general rule is :

z = a ( x + y) = a.x + a.y

This is the most useful "distributive law". We use this in many problems in algebra. Example: Poly High has four classes.In each class, we have students in the following numbers: 12 boys,14 girls. The numner of lockers for each student is 3. Find the total number of lockers needed.The cost of the lockers are different for boys lockers and girls lockers.Boy's locker cost $ 30 each while girl's locker costs 4 0 $ each.Work out the total cost. Total cost C = a x + b y where is a is th e number of boys ,X the cost of boys locker and b ,the number of girls and the cost of girls locker. C = 12 x 30 + 14 x 4 0 = 360 + 560 = 920 $ We can expand the distriubtive law for more than two terms. Suppose : z = x (a + b + c ) then z = ax + bx + cx C= Total number N = 3 ( 12 + 14 ) Here is a great example: John has a garden and lawn. The areas of the two pieces are x and y . He spends the garden labour cost as $A per square foot....Write an algebraic formula to calculate the labour cost for this work. The general formula is simply this: COST C = A ( x + y) = A.x + A.y Why should we write this formula? A formula like this is a general formula many can use.Whenyou calculate ,you put in the values of A,x and y. You get cost C for any particular case or specific garden. Calculate the cost if x = 2000 square feet and y = 4 00 square feet and A is just $1.5 per 100 square foot. C = (1.5/100 ).( 2000 + 4 00) = $36.00 Expanding factors:

Often we end up with the following formula: z = a ( x + y ) + b ( x + y) Note that (x + y ) occurs twice and is common to both the terms. then we write: z = (a + b) (x + y) If you expand you get: z = a (x +y) + b (x + y) = ax + ay + bx + b y. Example: Amy has a house which need painting the walls and fixing new tiles on the floor.The wall area is x and floor area is y. She also needs to do plastering the walls and the floor before painting.The cost of plastering per qsaure foot is 'a' The cost of painting is 'b' per square foot.Find the expression for total cost: Cost C =(a +b) (x + y) General formulas: There are a few general formulae or formulas widely used in algebra.Let us get three now from the previous example: Take : z = (a + b) ( x + y) = ax + ay +bx + by Now, put ,just for fun: a= x and b = y Then z = (x + y ) . (x + y) 2 2 2 = (x + y) = x + xy + yx + y 2 2 = x + 2 xy + y 2 2 2 Likewise ( x -y ) = X - 2xy + y We have another great formula, a very sueful one: (X+Y)(X-Y) = X (X - Y) + Y (X-Y) = X.X - XY + YX - Y.Y 2 2 =X - Y Let us see a few examples of uses of these formlas. Suppose you find a garden with pathway around it which is almost a square.The widht og the garden is 200 feet.The path way is 20 feet width on either side.The length of the garden is also 200 feet.The pathway runs around the garden...What is the total area of the garden? Area = length X width = (200 + 4 0).(200 + 4 0)= 200.200 + 2.200.4 0 + 4 0.4 0 ...Find the total.

Linear Equations Consider this simple problem. Jane buys on line 4 packets of pencils at $3 and packing and shipping charge is $7 for each shipment. We can write a simple equation: Bill amount = 3 .X + b where 3 is a constant or cost of each packet, while X is the number of packets you may buy.Shipping charge b is also constant . Now X=4 and b= 7 Therefore in general we write: Y = a X + b This is a linear equation. As X increases, in this case, y also increases.If X is doubled,Y is also doubled ,if b=0. If you know about plotting a graph, you will know that we mark x inthe horizontal axis ,while y is marked in the vertical axis. Then this equation results in a straight -line graph....well, that is why it is called a "linear equation or relation". You will note that a is called 'slope' and b is called the intercept.Recall these points here.! We gave a simple example of a bill being preapred.Linear equations are most widely used for many problems and business calculations...Why, because it si simple to use and in many situations ,it is quite good. Let us see a few examples. John makes school furniture. He has to pay the rent,elelcric bill and other expences for the factory...These expences add upto $5000 each month.In a month,he produces 100 sets of tables and chairs.For each set, he spends $ 50 for wood,nails,labor charges and so on. The total cost C = 100 .50 + 5000 We can write in general: C =n.X + b where n is the number produced, X,cost for each set and b is called the fixed cost [=5000] We take X as the variable. John sells each set for $80 or higher.If he sells only 90 sets in one month,his revenue is given by this equation: Revenue R = 80.y = s Y where s is the selling price. Note tht this is also a linear equation,but b =0 [intercept = 0] Now we can write another equation: Profit equation Profit = P = Revenue - cost = R - C = sy - (nX +b) = sY - nX - b In this example Profit P = 80.90 - 100.50 - 5000 = 7200 5000 - 5000= -2800.

Note here the profit is negative.What does this mean---no profit ,but a loss of $2800. Solving equations We need to 'solve' equations ---that is find the value of unknown variable,usually marked x ,y and so on.Let us take simple ones with only one variable X. Solve: 2 (x + 3) = 10 Expanding, we get 2.x + 6 = 10 Get x term in the left side while numbers are put in right side. 2.x +6 -6 = 10 - 6 2.x + 0 = 4 2.x = 4 x=2 Take another example: 2 (x + 3) + 4 = 3( x-2) - 3 First expand both sides, removing the brackets: 2.x + 6 + 4 = 3.x - 6 - 3 Simplify each side: 2.x + 10 = 3.x - 9 Next remove the x term from right side and the number from the left side: 2.x - 3.x + 10 = 3.x - 3.x - 9 -x + 10 = -9 -x + 10 -10 = -9 -10 - X = - 19 or X = 19 -----------------------------------------------------------------A simple example again: John has a rectangular garden with length of 4 00 feet and width of 150 feet.He has to fence the perimeter and the cost of fencing per foot is $22. Find the total cost.Let us set up the general equation: perimeter P = 2 ( L + W) where Lis the length and W,the width. Cost = C.P where c is th ecost per unit length or per foot Total Cost = C.P = 2c ( L + w) = 2 (22 )( 4 00 + 150 ) = 4 4 . 550 = $24 200. We shall see some more examples of solving equations: Suppose: 3x - 12 = 2x + 6 In htis example ,we ahve x terms on both sides.Let us have x term on the left side and the number term on the right.Subtract 2x from right side and left side: 3x - 2x - 12 = 2x - 2x + 6 x - 12= 6 Now take the number term from laft side to right side:[ I am adding 12 to both sides]. x -12 + 12 = 6 + 12

x = 18 We come across many types of espressions in algebra.Let us see a few of them. Inverse relation One common type is as follows: y=k/x where k is a constant. Let k= 12 Y = 10 / x As x increases, y decreases; if x = 2 ,we get y=6 if x=3 , y= 4 ; if x= 4 ,y=3 ; if x= 6 ,Y =2. Such a realtion is called "inverse relation"....In physics, you might have studied the gas law: P x V = constant K for a given temperature.or V = K/ P Direct relation is : y = k X or just a linear relation. The Newton's second law leads to : F = m x a where F is force, m,mass and a,the acceleration of abody. If acceleration a is doubled, F ,force is also doubled,keeping m,mass as constant. [In a rocket or aircraft, as fuel burns fast, m keeps changing!.Incars,yes the fuel mass is decreasing but slowly] If pressure P, is increased, V, volume is reduced. Power Law A useful algebraic equation is the power law: y = k x ^n Here x is raised to the power of n. assume that a baloon is a sphere: its volume V = (4 /3) . pi. (r)^3 Now V = k r^3 . If you double the radius of the baloon,its volume increases by 8 times. General Remarks At this stage, I should leave this short tutorial on algebra.We may add more stuff later...Remember, with this basic foundation in algebra, you should be able to learn more math and attempt several problems.