I warmly welcome every steemian in week 2 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 branch 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 applications.
Equations
If I talk about an equation then this is a statement which is used for expressing equality of two mathematical expressions with each other and an equation can contain variables and constants mostly and it is denoted by sign of equality (=).
Algebraic Example:
2x + 3 = 5
Practical Example:
Suppose there is a person whose name is Ali and he has $5 for spending at snacks and from these budget he purchased a sandwich of cost $2 and are drink of cost $1. Calculate that how much money is left by supposing that X is amount of money that is left by Ali.
Equation for this is represented as;
$2(Sandwich)+$1(Drink)+X(Money left)=5(Total money)
See that there is an equality at left hand side and right hand side of equal sign so its an equation which is representing equality of two mathematical expressions.
Inequalities
If I talk about inequality then any person can understand through this term that there is something inequal so it is defined as statement which is used for expressing relationship between two mathematical expressions which are not equal to each other so definitely one expression would be greater than other expression. If I talk about their representation then inequalities may be represented as <, >, ≤, ≥.
Algebraic Example:
2x + 3 > 5
Practical Example:
Suppose there is a book shelf which have maximum capacity of weight at around 50 pounds and if there one book whose weight is around 3 pounds then calculate that how much books could be placed at book shelf by supposing that x is representing number of books.The inequality is represented as 3x ≤ 50 (total weight capacity).
Types of equations
There are different types of equations from which today I am discussing linear and quadratic equation;
1. Linear Equations
If I talk about linear equations then this is a type of equation in which variable highest power can be one and in general form it can be written as;
ax + by = c
Here a, b, and c are constants, and x and y are variables.
Algebraic Example: 2x + 3 = 5
Practical Example:
Again suppose that Ali has $5 for spending at snacks and he bought sandwich of $2 and drink of $1. Now have to calculate that how much money he left?
2. Quadratic Equations
If I talk about quadratic equation then this is basically a type of equation in which highest power of variable can be 2 and generally it can be represented as;
ax^2 + bx + c = 0
Here a, b, and c are constants and x is a variable.
Algebraic Example:
x^2 + 4x + 4 = 0
Practical Example:
Suppose there is a ball which is thrown in upward direction from ground with 20m/s initial velocity and height of ball which is thrown above ground can be given by equation h(t) = -4.9t^2 + 20t in which h is representing height in metres and t is representing time in seconds.
Types of Inequalities
There are different types of inqualities from which I am discussing today about linear inequalities and compound inequalities.
1. Linear Inequalities
If I talk about linear inequality then this is a type of inequality in which variable highest power can be 1 and it can be represented as;
ax + by > c, ax + by < c, ax + by ≥ c, or ax + by ≤ c
Here a, b and c are constants and x and y are variables.
Algebraic Example:
2x + 3 > 5
Practical Example:
Suppose there is a bakery which is selling 200 minimum loaves of bread each day. If bakery is selling x loaves of bread, then they make a profit of $0.50 per loaf.
2. Compound Inequalities
If I talk about compound inequalities then this is a type of inequality which is used for combining two or more inequalities by the use of words "and" or "or". It can be representing as follows;
a ≤ x ≤ b or a ≥ x ≥ b
Here a, b and x are constants or variables.
Algebraic Example:
-3 ≤ 2x + 1 ≤ 5
Practical Example:
Suppose there is a company which is producing boxes of serial which weight should be in between 10.5 kg to 11.5 kg.
There are some of the methods along examples used for solution of linear and non linear equations and inequalities that are discussed one by one below;
Linear Equations and Inequalities
Method
1. Addition and Subtraction:
Addition and subtraction of same value to both sides of equation or inequality is one of the most common and easy method used for solving linear equations and inequalities.
2. Multiplication and Division:
Multiplication and division of both sides of equation or inequality by the use of same non zero value is another one of easy method for solving linear equations or inequalities.
Example 1: Solving a linear equation
Equation: 2x + 3 = 7
Solution:
• First step is to subtract 3 from both sides so it will be written as 2x=7-3
• Second step is its simplification so after simplification it is 2x=4.
• Third step is to divide both sides by 2 so after solving it is x=4/2.
• Last step is simplification for getting and a result so after simplifying it is x=2.
Example 2: Solving a linear inequality
Inequality: 3x - 2 > 5
Solution:
• First step is addition of 2 at both sides so it can be written as 3x > 5 + 2
• Second step is simplification so after simplification it can be written as 3x > 7
• Third step is to divide both sides with 3 so after that it can be written as x > 7/3
• Last step is simplification so after simplification final outcome is x > 2.33
Non-Linear Equations and Inequalities
Method
1. Factoring:
If I talk about factoring then this is a method to factor an equation or inequality in products of two binomials.
(Binomial is a mathematical expression that consist of two terms and these terms are separated by plus or minus sign e.g,a+b)
2. Quadratic Formula:
If I talk about quadratic formula then this is again another one of the most important method which use quadratic formula for solution of quadratic equations.
3. Graphical Method:
If I talk about graphical method then again this is most important method for solution of non linear equations and inequalities which involves graphing of equation or inequality at coordinate plane and then finding the solution.
Example 1: Solving a quadratic equation
Equation: x^2 + 4x + 4 = 0
Solution:
• First step is to factorize this equation as (x + 2)(x + 2) = 0
• Second step is to solve for X so it can be solved as x + 2 = 0 --> x = -2
Example 2: Solving a non-linear inequality
Inequality: x^2 - 4 > 0
Solution:
• First step is to factorize this inequality as (x - 2)(x + 2) > 0
• Second step is important and it is about solving for x which can be solved as x - 2 > 0 and x + 2 > 0 --> x > 2 and x > -2
• 3rd step is all about combining the solutions for writing it properly as final outcome x < -2 or x > 2
There are some algebraic methods for solving equations and inequalities that I am describing below;
Algebraic methods for solving equations
1. Substitution Method:
Substitution method is one of the most common method for solving equations in which we have to substitute a value or an expression for variable present in an equation
Example: Solve this equation 2x + 5 = 11
• First step is of subtracting five from both sides and then we will get 2x = 11 - 5
• Second step is of simplification after that we get 2x = 6
• Third step is of dividing both sides with 2 after that we get x = 6/2
• Lastly with more simplification finally we get x = 3
1. Elimination Method:
If I talk about elimination method then this is also important for solving equations in which we eliminate one variable from system of equations.
Example: Solve system of equations;
x + y = 4
x - y = 2
• First step is all about adding two equations after that we get 2x = 6
• After that in second step we divide both sides with 2 and finally we get x = 3
• In third step we will substitute X into one of original equations and it will be written as 3 + y = 4
• Lastly we subtract 3 from both sides for getting final outcome which is y = 1
Algebraic methods for solving inequalities
1. Addition and Subtraction Method:
If I talk about addition and subtraction method for solving inequalities then it includes adding and subtracting similar values at both sides of inequality.
Example: Solve inequality 2x + 3 > 5
• First I have subtracted 3 from both sides after that we get 2x > 5 - 3.
• Secondly we have simplified it for getting 2x > 2
• Lastly we will divide both sides with 2 for getting x > 1
1. Multiplication and Division Method:
If I talk about multiplication and division method then it includes multiplication or division of both sides of inequality with same non zero value and it is used for solving inequality.
Example: Solve inequality 4x < 12
• First we will divide both sides with 4 for getting x < 12/4.
• Lastly we will simplify it for getting x < 3.
If I talk about use of calculators then this is most effective way for solving equations or inequalities particularly if you are dealing with complex or large numbers. Below I am explaining some of the calculators that are used for solution of equations or inequalities.
Types of calculators
1. Graphing calculators:
If I talk about graphing calculators then these are useful for graphing functions, solving equations and inqualities.
As an example;
Texas Instruments, calculators and Casio calculators.
2. Scientific calculators:
If I talk about scientific calculators then these are used for performing advanced mathematical operations which include solving equations and inequalities.
As an example;
HP calculators and TI scientific calculators.
3. Online Calculators:
If I talk about online calculators then these are available online which are used for solving equations and inequalities.
As an example;
Wolfram Alpha,Symbolab and Mathway
Steps to use calculators
Following are the steps of using calculators;
• First step is of typing equation or inequality in calculator by the use of correct Syntax and formatting.
• Second step is of selecting solving function at calculator that can be labelled as "Solve","Equation" or "Inequality"
• Third step is to identify variable in equation or inequality and specifying it at calculator.
• Fourth step is pressing enter and execution button for solution of equation or inequality.
• 5th step is to check solution that is provided by calculator for checking its accuracy.
| • | Task 1 |
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• Explain difference between linear and quadratic equations. Provide examples of each type of equation and describe their general forms.
| • | Task 2 |
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• Describe two different types of inequalities(Which are not explained in course). Provide examples of each type of inequality and explain how to solve them.
| • | Task 3 |
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• Solve the following linear equation: 2x + 5 = 11.Show step-by-step solution and share its practical example of how this equation can be applied in real life scenario.
(You are required to solve this problem 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 |
|---|
Scenario: Tom's Bakery
Tom's bakery is a most famous bakery of tow and it's famous for its yummiest pastries and bread. Tom which is owner want for making sure that he has sufficient ingredients for meeting needs for his famous chocolate cake and chocolate cake recipe needs 2 cups of flour, 1 cup of sugar and 1/2 cup of cocoa powder for each cake.
| Equation | If Tom wants for making x cakes and he has 10 cups of flour, 8 cups of sugar and 4 cups of cocoa powder then calculate number of cakes that can be made by Tom? |
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| Inequality | Tom wants for making sure that he has sufficient sugar for meeting needs for his chocolate cake. He also knows that each cake needs 1 cup of sugar and he has 8 cups of sugar which are available. He also knows that he wants to make at least 6 cakes. |
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Solve for equations and inequality both from above scenario
• Scenerio: Number of gallons Ashley needs.
Suppose there's a person named Ashley who is planning for a road trip from City A to City B.If I talk about distance between two cities then it's around 240 miles. Ashley's car fuel tank have ability of holding 12 gallons of Suppose that his car gets 20 miles for each gallon then how many gallons of gas may Ashley purchase if he already has 2 gallons in his tank and he wants to have minimum 1 gallon left over when he arrives at City B?
| Equation | Let's consider x as number of gallons of gas Ashley can purchase. |
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| Inequality | x + 2 ≥ (240/20) - 1 |
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You have to solve for inequality now for finding minimum number of gallons of gas Ashley required to purchase.
(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)
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