SLC S22/W3 : Equations and Systems of Equations

sheikhtuhin (61)in #algebra-s22w3 • 3 days ago

Hello friends, I hope everyone is well? Alhamdulillah I am fine.
I am @sheikhtuhin
From #Bangladesh

Dear friends, today I am going to join SLC S22W3//Equations and Systems of equations.This learning challenge is organized by the beloved teacher of this platform, @khursheedanwar Sir.I am very grateful to Sir because now I don't need to practice Maths. I have forgotten it for a long time, but thanks to Sir's arrangement, I am able to try those old topics again.

_{edit by inshot}

Without further ado, I'll start today's homework right away.

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

Difference between linear equation system and non-linear equation:

Linear Equation System	Non-Linear Equation
🔲 In a system of linear equations, each equation represents a straight line. Here, the powers of the variables are always 1 and there are no products of the variables.	🔲A nonlinear system of equations has at least one equation with a power greater than 1, a product of variables, or a nonlinear function.
🔲The solution method is relatively simple.	🔲 The solution method is relatively complex.
🔲Each equation represents a straight line, and the solutions are the point or points where the lines intersect.	🔲 Each equation represents a curve, and the solution is the point where the curves intersect.
🔲The solution is by substitution method, addition-subtraction method, and matrix or Gaussian elimination method.	🔲 Solved by graphical method, Newton's method, numerical analysis method.
🔲General form:- a1X1+a2X2+.....+anXn=b, (Where, a1, a2 and an are constants and X1,X2, Xn is variable).	🔲 General form: f(X1,X2,.......Xn)=0
🔲 For example, example: 2x+3y=6 , x-y=2	🔲For example, x²+y²= 25 , xy=12

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)

Apart from the above issues, equations can be solved using Cramer's rule.It is possible to solve this with the help of matrices and determinants.My mobile doesn't support the markdown of determinants and matrices, so I'll try to solve it in a simple way.I will try to explain Cramer's rule in a general way using matrices and determinants.

To solve using Cramer's rule, there are certain conditions:

1. The number of equations and the number of variables must be equal.

####2. The determinant of the coefficient matrix cannot be zero.

Steps:

If the two equations,

a1X+b1Y=c1......(i)

& a2x+b2y=c2.......(ii) It is.

Then according to Cramer's rule,

We need to find the determinant of the coefficient matrix.

D= |a1 b1|
|a2 b2|

= (a1b2 - a2b1)

We need to find the determinant for x:

Dx = |c1 b1|
|c2 b2|

  = (c1b2 - c2b1)

We need to find the determinant for y:

Dy= |a1 c1|
|a2 c2|

  =( a1c2 - a2c1)

Determine the values of the variables:

X= Dx/D

Y= Dy/D

This is basically how it is possible to solve equations using Cramer's rule, but in this case, it is important to adhere to the above conditions.

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

Solution

I will find the values of X and Y by solving the equations given above using Cramer's rule.

(a) For this.

Given,
X+2y= 7 ..........(i)
3x-2y= 5.........(ii)

Step-1

For these two equations,

D=
|1 2|
|3 -2|

 = 1×(-2) - 3×2
 = -2-6
 = -8

Here ,
The determinants will be Dx for x and Dy for y.

Dx=
|7 2|
|5 -2|

    =7×(-2) - 5×2
    = -14-10
    = -24

Dy=
|1 7|
|3 5|

    = 1×5 - 3×7
    = 5-21
    = -16

Second step:

X = Dx/D
= -24/-8
= 3

Y = Dy/D
= -16/-8
= 2

Result:

In case of (a) (X,Y)= (3,2)

In the case of (b),

What is in the equation is two,

4x+6y =2..........(i)
X-2y  = 3...........(ii)

Step-1

For these two equations,

D =
|4 6|
|1 -2|

  = 4×(-2) - 1×6
  = -8-6
  = -14

Here ,
The determinants will be Dx for x and Dy for y.

Dx=
|2 6|
|3 -2|

    = 2×(-2) - 3×6
    = -4 -18
    = -22

Dy=
|4 2|
|1 3|

    = 4×3 - 1×2
    = 12-2
    = 10

Second step:

X = Dx/D
= -22/-14
= 11/7

Y = Dy/D
= 10/-14
= -5/7

Result:

In case of (b) (X,Y)= (11/7, -5/7)

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)

Solution:

Given,
Company "A" wants to produce 50 units.
That is, X= 50.

Equations:
2x+3y=130.........(i)
X+2y=110..........(ii)

Step-1

(i) Putting x=50 in equation no., we get

 2x+3y =130

=> 2×50+3y =130
=> 100+3y =130
=> 100-100+3y = 130-100

_{[Subtract 100 from both sides]}
=> 3y= 30
=> 3y/3= 30/3 _{[Divide both sides by 3]}
=> y= 10

Step 2

Put the values of "x" and "y" in the second equation.

   x+6y=110
=>50+6×10=110
=> 50+60= 110
=> 110= 110

Final result:

If Company "A" wants to produce 50 units, then Company "B" can produce 10 units.

Task-4 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?

Given,

Equation two:

   X+2y =80......(i)
  2x+y = 70.......(i)

The question says that he wants to make 30 vanilla cakes.Then x= 30.

Step-1

(i) We get the value of x by substituting it into equation no.

    X+2y = 80
 => 30+2y= 80
 => 2y = 80-30
 => 2y = 50
 => y = 50/ 2
 => y = 25

Step-2

Substituting the values of x=30 and y=25 is a truth check and a truth test for equation number two.

 2x+y=70

=>2×30 + 25 = 70
=> 60+25 ≠70
=> 75≠70
The equation is inconsistent.

Decision making:

Since the equation was checked for correctness, it was found that one is not consistent with the other.Therefore, it is not possible to determine the value from this equation in this way and it is not correct but incorrect.

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I invite my friends to join me in this learning challenge.I hope you can practice your mathematical life again through this learning challenge, even after you've gone through it.@kibreay001 @ariful2 @pea07 @abdulmomin

Thank you very much everyone

Dear @khursheedanwar Sir, I did not understand question number 4 correctly today, so I tried to give an idea of the answer to this question based on what I understood.