I warmly welcome every steemian in week 1 course about algebra!
It's important to discuss introduction of algebra and important terminologies about algebra before getting involved into in depth of main topic that I am teaching you guys today!
Algebra is basically a major bench of mathematics which consists of study of variables and their relationships with eachother.If I talk about algebra then It includes usage of different symbols, equations, and formulas for solving problems and for modeling real world situations.
Variables
If I talk about variables in algebra then variable is a symbol which is used for representation of a value which can alter, modified or changed.
How variables can be represented?
Variables can be presented by using any alphabetical letters like x, y or z.
2x+2y
X and Y are variables.
Expressions
If I talk about expression then it is a group of variables, constants and mathematical operations which combine together.
How expressions can be represented?
If I talk about representation of expressions then they can be represented by including variables, constants and mathematical operations like;
2x+3
X is a variable because it can change
2,3 are constants because these values will not change.
(+) sign is a mathematical operation.
Real world problem and it's solution
It is required to define a variable X as number of hours worked in one day.Now write an expression for total cost earned in one day in which you can consider that $15 is hourly wage.
In solution,X is number of working hours in one day(It's a variable that can be changed).In an expression it will be written as 15X which is total amount earned in one day.
Following are types of variables and expressions!
Types of variables
Independent variable
If I talk about independent variable then are those values which don't depend on other variables like in real world when you have to determine exercise effect at loss of body weight then amount of exercise that you can perform is independent variable.
Suppose there's an equation Y=2x+3,then x is independent variable in this equation because it is not depending on other variables.
Dependent variable
If I talk about dependent variables then these are those variables which is depending on other variables.In real world example when you are determining exercise effect for weight loss of body then weight loss is depending on amount of exercise so weight loss is depending that's why it's an dependent variable.
Suppose there's an equation which is Y=2x+3 then Y is depending on other variables so it's dependent variable.
Types of expressions
Numerical expressions
If I talk about numerical expression then it consists of just numbers and mathematical operations like if there's are costs of 2 tickets for a movie are $5+$7=$12.
Suppose 2×3+4=10 then 2,3,4 are numbers and (×)(+) are mathematical operations.
Algebric expressions
If I talk about algebric expression then it consists of variables,constants as well as mathematical operations like expenses of x tickets for a movie is $10x+$5.
Suppose 2x+3 as an expression so it contains X as variable,2 and 3 as constants and (+) as mathematical operation.
What is PEMDAS?
PEMDAS are mnemonics which are helpful for people in remembering order of operations in mathematics. PEMDAS stands for:
• P stands for parentheses which are useful for evaluation of expressions in parentheses as first order.
• E stands for exponents which are used for evaluation of any exponential expressions as second order like 2^3.
• M stands for multiplication in third order.
• D stands for division in fourth order.
• A stands for addition as fifth order.
• S stands for subtraction as sixth oder.
Why is PEMDAS used?
PEMDAS is useful for providing a standard way for evaluation of mathematical expressions. Without PEMDAS rule, expressions that have multiple operations may be interpreted in different ways that creates more confusions and errors in end result.
How is PEMDAS used?
For using PEMDAS there's a need of simplification for following order of operations which are highlighted or outlined Evaluation of any expression should be started from parentheses, then to exponents, multiplication and division and finally addition and subtraction as I have stated above also order wise!
Using PEMDAS let's evaluate following expression
2×3+10-5 is an imaginary expression so let's solve it by using PEMDAS
Parentheses,Exponents and division are not applicable to expression which I taking as an example so;
• First there's a need to multiply 2 and 3 so this is 6 after multiplication.
• Secondly there's a need of adding 10 into 6(obtained after multiplication)so as a result this is 16 after adding 10 into 6.
• Thirdly there's a need of subtracting 5 from 16(obtained after calculation) so it's 11 as a result when I minus 5 from 16 as 16-5=11.
Finally expression value we got after applying PEMDAS is 11
Variables and expressions are applied in real world problems as follows;
Cost calculation scenario
There's a company which is producing t-shirts at expenses of $5 per shirt as well as a fixed cost of $100 was also added in it already for one day.x is representing number of shirts(these are variable) which are produced per day so total cost may be represented by following expression;
Total Cost = 5x + 100
This expression may be useful for calculating total costs for production of particular number of shirts.
Distance calculation scenario
If there's a car travelling from one city A to another city B at a average speed of 60 miles for one hour. If x is representing time traveled in hours then distance traveled may be represented by following expression;
Distance = 60x
This expression may be useful for calculating distance traveled by car.
Business problem
There's a company which is selling a product for $10 per unit and cost of production for each unit is $5. If x is representing number of units which are sold so profit may be represented by following expression;
Profit = 10x - 5x
This expression may be useful for calculating profit which is earned by company after sold out of particular number of units.
Now these are variables and expressions which are identified in above examples;
Cost calculation
Distance calculation
Business problem
Simplifying algebraic expressions
Simplifying algebraic expressions consist of combining similar terms and then removing parentheses and performing other operations for making expression more manageable.
Example number 1;
Suppose there's an expression which is 2x+3x-4
For simplification of expressions you need to combine similar terms together like;
2x+3x which is equal to 5x after adding them.Now expression after simplification can be written as follows;
5x-4
Example number 2;
Here is an imaginary expression 3(2x+1)-2x which need to be simplified so first there's a need of distributing 3 to terms which are inside parentheses or brackets.Now it will be written as follows;
3(2x + 1) = 6x + 3
There's a need of combining similar terms together now;
6x+3-2x=4x+3
4x is extracted by subtraction of 6x-2x
Finally simplified expression is 4x+3
Solving algebraic equations
If you have to solve algebraic equations then it includes to isolate variable (usually x or y) on one side of the equation.So now it's a time to solve algebric expressions by using following examples;
Example number 1;
Suppose 2x+3=7 is an equation and we want to solve for value to X so we need to subtract 3 from both sides for isolation of variable at one side of equation.
It will be written as follows then;
2x=7-3
2x=4
Now 2x are in multiplication form and when for getting value of x 2 will be moved on other side then it will be represented as follows;
x = 4/2
x = 2
Finally value of x is 2 after solving this algebric expression
Example number 2;
If there's an equation x/2 + 3 = 5 and we want to solve it then following step by step approach will be followed;
We need to isolate variable at one side of equation so for this 3 will be subtracted from both sides;
x/2=5-3
x/2=2
You can see that above 2 is getting divided with X so after it move to another side it will get multiplied with constant already present there so it will be solved as follows;
x = 2 × 2
x = 4
Finally 4 is final value of x after solving algebric expression
| • | Task 1 |
|---|
• Explain any two variables and expressions types other than that which are explained in this course.(Practical and algebric examples are required!)
| • | Task 2 |
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• Show your way of evaluating of an algebraic expression if values of variables are given? Step by step explanation required!
(The more you will be detailed and accurate,the more your task will be perfect!)
| • | Task 3 |
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• Simplify this expression: 3(2x - 1) + 2(x + 4) - 5
• Evaluate this expression: (x^2 + 2x - 3) / (x + 1) when x = 2
• Solve the following equation: 2x + 5 = 3(x - 2) + 1
(You are required to solve these problems at paper and these share clear photographs for adding a touch of your creativity and personal effort which should be marked with your username)
| • | Task 4 |
|---|
• Suppose there's a bakery selling a total of 250 loaves of bread per day. They are selling whole wheat and white bread loaves with numbers of whole wheat loaves sold being 30 more than the number of white bread loaves. If x is representing number of white bread loaves sold out and bakery is making a profit of $0.50 for each white bread loaf and $0.75 for each whole wheat loaf then please write an expression for representing bakery total daily profit.
• Suppose that cost of renting a car for a day is re-presented by the expression 2x + 15 and here x is the number of hours in which car is rented. If the rental company offers a package of 3x - 2 dollars for customers who take car at rent for more than 4 hours then write an expression for the total cost of renting the car for x hours and show how you simplify it.
(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)
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