SLC S22W3//Equations and Systems of equations

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Greetings

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I warmly welcome every steemian in week 3 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.

Defining equations and systems of equations

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.

Systems of Equations

If I talk about system of equations then it is a collection of two or more equations which are containing multiple variables.Primary aim for system of equations is for finding values of variables which are used for satisfaction of all equations in a simultaneous way.

Types of Systems of Equations

Linear Systems

If I talk about linear systems then these are those equations which contains highest power of the variables and it is 1.

Nonlinear Systems

If I talk about non linear equations then these are those which have highest power of variables which is greater than 1.

Practical Example

Suppose Tom have $100 for spending at tickets at concert.Tickets total cost us around $20 for adults and $10 for children. If Tom is buying 2 adult tickets and some tickets for children and spending exactly $100 then how much tickets for children Tom can buy?

Suppose A be is number of adult tickets and C is number of children's tickets.

Equation 1 can be written as;

20A + 10C = 100 (Total cost)

Equation 2 can be written as

A = 2 (Number of adult tickets)

Algebraic Example

Solving system of linear equations by following steps;

2x + 3y = 7
x - 2y = -3

For solving this system, we may use substitution or elimination methodologies.

In practical example I am substituting A = 2 into Equation 1 and then I am solving for C;

20(2) + 10C = 100
40 + 10C = 100
10C = 60

C = 6

Tom may purchase 6 tickets for children.

If I talk about algebraic example then we may also use elimination method for solving for x and y.

First there's a need for multiplying equation 2 by 2 and then adding it to equation 1 for eliminating y-variable so;

2x + 3y = 7 2x - 4y = -6
4x - y = 1

Now we are solving for x and y by the use of substitution or further elimination methodology.

Solving linear and non linear systems of equations

Linear Systems

If I talk about linear systems then they may be solved by use of substitution, elimination, or graphical methodologies.

Algebraic Example

For solving systems we need to follow the steps below;

2x + 3y = 7
x - 2y = -3

If I talk about use of elimination method, then we need for multiplying Equation 2 by 2 and then add it in to Equation 1 as I am doing below;

2x + 3y = 7 2x - 4y = -6

• y = 1

y = -1

This step is for substituting y into Equation 1 as;

2x + 3(-1) = 7
2x - 3 = 7
2x = 10
x = 5

Practical Example

If a bakery is selling total of 250 loaves of bread and buns each day and they are selling a combination of whole wheat and white bread. If they are selling 30 more whole wheat bread in comparison to white bread and they are selling 120 whole wheat bread then how much white bread they can sell?

Suppose that x is number of white bread.

Now equations can written as;

Equation 1: x + (x + 30) = 250 (Total bread)
Equation 2: x + 30 = 120 (Whole wheat bread)

If we have to solve Equation 2 for x then it gives x = 90.

Nonlinear Systems

If I talk about nonlinear systems then it may be solved by the use of substitution, elimination, or numerical methodology.

Algebraic Example

For solve system we need to follow following steps;

x^2 + y^2 = 25
x + y = 5

By the use of substitution I am solving for Equation 2 for y;

y = 5 - x

Now it's a step for substituting y in Equation 1 as follows;

x^2 + (5 - x)^2 = 25

Now there's a need for expanding and solving for x.

Practical Example

Suppose profit of company is impacted by number of units sold (x) and price per unit (y). The profit is modeled by this equation which is P = 2x^2y - 3xy^2. If profit of company is $1000 and they are selling 10 units then calculate for price per unit?

Suppose x = 10 and P = 1000

Equation 1: 1000 = 2(10)^2y - 3(10)y^2

Now you can solve for y by use of numerical method or graphical method.

Applying substitution, elimination and graphical method

If I talk about solving systems of linear equations then there are three methods I am going to teach you guys today which are substitution, elimination and graphical method.

Substitution Method

Solve one equation for one variable

If I talk about step 1 then you need to choose one equation and solve for one variable in terms of the other variable or variables.

Substitute into the other equation

If I talk about second step then it is used for substitutions of expression from step 1 in other equation.

Solve for the other variable

If I talk about step 3 then it is used for solving resulting equation for other variable.

Example

For solving system we may follow following steps;

2x + 3y = 7
x - 2y = -3

• First solve Equation number 2 for x: x = -3 + 2y

• Secondly we need to substitute in Equation 1 as : 2(-3 + 2y) + 3y = 7

• Now we need to solve for y as y = -1

Elimination Method

Make coefficients same

If I talk about step 1 then there's a need of multiplication of both equations by necessary multiples like coefficients of one variable like x are same.

Add or subtract equations

In this step there's a need of adding or subtracting equations for eliminating one variable.

Solve for the other variable

For solving resulting equation for other variable.

Example

For solving system we need to follow following steps;

2x + 3y = 7
x - 2y = -3

• In step 1 we need for multiplication of Equation 2 by 2 as 2x - 4y = -6

• In step 2, there's a need of adding Equation 1 and modified Equation 2 as follows (2x + 3y) + (2x - 4y) = 7 + (-6)

• In step 3 there's a need for solving for x as x = 1/2

Graphical Method

Graph both equations

If I step 1 then there's a need of graphing both equations at same plane of coordinate.

Find the intersection point

In step two there's a need of Identifying point where two lines are intersecting.

Read the solution

If I talk about this step then there's a need of reading values of x and y from intersection point.

Example

For solving system we need to follow these steps;

2x + 3y = 7
x - 2y = -3

• In step 1 there's a need of graphing both equations.

• In step two there's a need of finding intersection point as (-1, 3)

• Now there's a need of reading solution as x = -1, y = 3

These methods may b helpful in solving systems of linear equations.If I talk about substitution and elimination methods then mostly these are most efficient but graphical methods are used for providing visual representation of solution.

Analyzing and interpreting solutions to systems of equations.

If I talk about analyzing and interpreting solutions for systems of equations then it includes to know the meaning of solutions in context of problem.So here I am explaining some steps for following;

Analyzing Solutions

Checking solutions

If I talk about step 1 then it's about verification that solutions are satisfying both equations in system.

Identify type of solution

If I talk about step two then it's about determining that if solution is single point which is (unique solution), or a line or curve which is (infinite solutions), or no solution which is (inconsistent system).

Considering context

In this step there's a need to find out abot context of problem that whether solution is making sense or not.

Interpreting Solutions

Understanding variables

In step there's a need of recalling that what a variable is representing in the problem.

Explaining solution

Is this step there's a need of describing solution in words by the use of context of the problem.

Check for reasonableness

If I talk about this step then it's used for ensuring solution which is reasonable and realistic.

Types of Solutions

I am going to share three types of solutions with you guys;

Unique solution

If I talk about unique solution then it's a single point used for satisfying both equations.

Infinite solutions

If I talk about infinite solutions then these are line or curve which are satisfying both equations.

No solution

If I talk about no solution then this type of system is inconsistent that's why in this case there is no solution.

Example

Imaging that there's a system of equations which is modeling cost of renting a car as follows;

2x + 3y = 120
x - 2y = -20

Here x is number of days for which car is rented and y is number of miles for which car is driven.

Solution

For solving system, we get x = 40 and y = 20.

Analysis and Interpretation

The solution (40, 20) is satisfying both equations. In context of this problem its meaning is for renting car for 40 days and driving car for 20 miles and here you can also see how this solution is reasonable and realistic.

By following these steps you may analyze and interpret solutions to systems of equations which can give surety of your understanding regarding meaning and context of the solutions.

Real world Application

Imagine that there's a bakery selling total number of 250 loaves of bread and buns for each day.They are also selling a combination of whole wheat and white bread. If they are selling 30 more whole wheat bread in comparison with white bread and they are selling 120 whole wheat bread then calculate amount of white bread they sell?

Suppose x as number of white bread.

Equations:

x + (x + 30) = 250
x + 30 = 120

Solution: x = 90

Analysis and Interpretation

If I analyse and interpret solution then it means that bakery is selling 90 white bread and 120 whole wheat bread for each day.

Modeling of real-world problems using systems of equations.

If I talk about modeling of real world problems then systems of equations are used which involves translating of problem in a mathematical representation, solving of system, and interpretation of results.I am sharing at end of course some real world problems with you guys by use of system of equations.

1. Cost Analysis

Suppose there's a company producing two products which are A and B.Now if I talk about cost of production x units of A and y units of B is given by system then it may be written as follows;

2x + 3y = 120 (cost of materials)
x + 2y = 100 (cost of labor)

2. Nutrition and Diet

Suppose there's a person needs daily intake of 2000 calories and 50g of proteins.If there are two foods as X and Y,then for providing following nutritional information for each serving we may write as;

Food X: 300 calories, 10g protein
Food Y: 400 calories, 20g protein

So this system is representative of daily nutritional which is as follows;

3x + 4y = 2000 (calories)
x + 2y = 50 (protein)

3. Economics and Finance

Suppose there's a person investing $1000 in two accounts and from them one is earning 5% interest and other is earning 7% interest.If I talk about total interest earning after one year then its $60 so now system will represent investment as follows;

0.05x + 0.07y = 60 (total interest)
x + y = 1000 (total investment)

4. Environmental Science

Suppose there's a lake having two sources of pollution, industrial waste (x) and agricultural runoff (y).If I say that total amount of pollution is 1000 kg then system which is representing pollution may be written as follows;

2x + 3y = 1000 (total pollution)
x + y = 500 (total waste)

5. Logistics and Transportation

Suppose there's a delivery truck travelling from city A to city B and then to city C.If I say that distance between A and B is 200 miles an distance between B and C is 300 miles so obviously total distance traveled will be around 500 miles so system which is representing distance may be written as follows;

x + y = 200 (distance A to B)
y + z = 300 (distance B to C)
x + y + z = 500 (total distance)

So these examples are illustrating how systems of equations may be useful for modelling real world problems in different fields. By solution of system we may get insights and may make informed decisions.

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Homework tasks

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•	Task 1

• Explain difference between linear and non linear systems of equations. Provide examples of each type of system of equation and describe their general forms.

•	Task 2

• Describe any one method for solving system of linear equations and share atleast one step by step algebric example.

(It should be other than substitution, elimination and graphing method)

•	Task 3

• You need for solving following system of linear equations:

(a)

x + 2y = 7
3x - 2y = 5

(b)

4x + 6y = 2
x - 2y = 3

(You are required to solve these problems at paper and then share clear photographs for adding a touch of your creativity and personal effort which should be marked with your username)

•	Task 4

Scenario number 1

Suppose there's a company producing two products, A and B.If cost of producing x units of A and y units of B is given by system then;

2x + 3y = 130 (cost of materials)
x + 2y = 110 (cost of labor)

If company wants for producing 50 units of product A then calculate how much units of product B they may produce?

(Solve the above scenerio based questions and share step by step that how you reach to your final outcome)

Scenario number 2

• Suppose there's a bakery producing two types of cakes which are vanilla and chocolate.If cost of producing x cakes of vanilla and y cakes of chocolate is given by system then;

x + 2y = 80 (cost of ingredients)
2x + y = 70 (cost of labor)

If bakery wants for producing 30 cakes of vanilla then calculate how much cakes of chocolate can they produce?

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Evaluation criteria

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Thanks

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