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akhl

@akhl

Dazzler

Posted: 15 years ago

Edit: Please post on what topics in Maths do you want a tutorial. 😊

1. Solving simultaneous linear equations in two variables

a. Method of elimination

b. Method of substitution

Edited by akhl - 15 years ago

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akhl

@akhl

Dazzler

Posted: 15 years ago

1. Solving simultaneous linear equations in two variables

You are given two equations in two variables. You have to find the values of variables such that those values satisfy both the equations. Then you have solved the equations simultaneously.

a. Method of elimination

The steps in the method of elimination are

i. Multiplied one or both equations by some number. Add the equations are added. The multipliers are chosen in such a way that on adding, one of the variable gets eliminated.

2. On adding you will get an equation in only one variable. Solve the equation to get the value of that variable.

3. Substitute the value of that variable into any one of the two given equations. Solve the equation to get the value of the other variable.

Let us take some examples:

Example 1.a.1

x + y = 10 -------(1)

x - y = 8--------(2)

Solution: Add the equations

x + y = 10

x - y = 8

__________

2x = 18 (on adding)

x = 18/2

x = 9

Substitute x = 9 into equation (1):

9 + y = 10

y = 10 - 9

y = 1

Ans: x = 9, y = 1

Example 1.a.2

2x + y = 7---------(1)

x + y = 5---------(2)

Solution:

Add equation (1) to (-1) * equation (2):

2x + y = 7---------(1)

-x - y = -5---------(-1) * (2)

_________________

x = 2

Substitute x = 12 into equation (2):

2 + y = 5

y = 5 - 2 = 3

Ans: x = 2, y = 3

Example 1.a.3

2x + 3y = 8

3x + 2y = 7

Solution:

Add equation (1)*3 to equation (2) * -2:

6x + 9y = 24 --------(1)* 3

-6x - 4y = -14---------(2)* -2

_______________________

5y = 10

y = 10/5 = 2

Substitute y = 2 into equation (1):

2x + 3*2 = 8

2x + 6 = 8

2x = 8-6

2x = 2

x = 2/2 = 1

Ans: x = 1, y = 2

Now that we have seen some examples, let us go over some tips on how to apply the method of elimination to solve simultaneous linear equations in two variables.

I will post that some time tomorrow.

Edited by akhl - 15 years ago

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akhl

@akhl

Dazzler

Posted: 15 years ago

a.1: Some tips on deciding what numbers to multiply the equations with

a.1.1: The coefficient of any one variable is same in both the equations.

Subtract one equation from another.

Example a.1.1.1:

x + y = 4

2x + y = 7

In this coefficient of y in each equation is +1. Subtract one equation from another.

Example a.1.1.2:

2x + 3y = 7

x + 3y = 5

In this coefficient of y in each equation is +3. Subtract one equation from another.

a.1.2: The coefficients of any one variable have opposite signs in the two equations but their absolute values are equal.

Add the equations.

Example a.1.2.1:

x + y = 4

2x - y = 5

In this coefficient of y in one equation is +1 and in the other it is -1. Add the equations.

Example a.1.2.2:

2x - 3y = 1

x + 3y = 5

In this coefficient of y in one equation is -3 and in the other equation it is + 3. Add the equations.

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akhl

@akhl

Dazzler

Posted: 15 years ago

Do any of you need any clarification in the meanings of the following terms?

1 Exponent

2. terms of an expression

2. Polynomial

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akhl

@akhl

Dazzler

Posted: 15 years ago

Cramer's Rule

Cramer's rule can be used to solve a system of linear equations if a solution exists. It can also tell if there is a unique solution set, no solution, or infinitely many solution sets.

Consider the linear system of equations:

ax + by = e

cx + dy = f

It can be written in matrix form as

So, write the coefficients in the first matrix called coefficient matrix. The coefficients appear in the same position in the matrix as in the given equations. Multiply it with a column vector which contains the variables. On the rhs, there is results column vector containing the rhs of the given equations.

To solve the equations using Cramer's rule find the following determinants:

There is a simple way to remember these determinants. The determinant D is the determinant of the coefficient matrix. To get the determinant of Dx, replace the elements of x column in D by the elements of the result column vector. To get the determinant of Dy, replace the elements of y column in D by the elements of the result column vector. The elements in the result column vector are e ad f. The variable x is the first variable. Therefore in Dx, the first column contains e and f. The variable y is the second variable. Therefore in Dy, the second column contains e and f. In general, you can find the determinant for any variable by replacing the column for that variable in the determinant of the coefficient matrix by the elements in the result column vector.

The rules for 3 equations in 3 variables are the same. Consider

ax + by + cz = j

dx + ey + fz = k

gx + hy + iz = l

The above set of equations can be written in matrix form as

Find the following determinants:-

As explained in the case of equations in two variables, D is the determinant of the coefficient matrix. Dx is obtained by replacing x-column in D by the elements of the result column. Likewise, Dy is obtained by replacing y-column in D by the elements of the result column, and Dz is obtained by replacing z-column in D by the elements of the result column.

Let us consider different cases:-

1. All of the determinants are non-zero: There is a unique solution set, i.e., there is exactly one set of values of the variables which will satisfy the given equations. Find the solution using the formulas:

x = Dx/D, y = Dy/D

For three variables, z = Dz/D

2. All of the determinants are zero: There are infinitely many solution sets, i.e., more than one set of the values of the variables will satisfy the given equations.

When there are infinitely many solution sets, then the equations are said to be indeterminate.

3. D = 0 and at least one of the other determinants is not zero: There is no solution set, i.e., no values of the variables will satisfy all the equations.

When there is no solution set, then the equations are said to be inconsistent or incompatible.

The steps explained in this tutorial can be extended to 4 or more variables.