Solving by elimination method

The elimination method is one of prominent methods in solution to the system of linear equations with two variables. The elimination method for solving equations may be apparent in certain cases where, the coefficients of a particular variable or variables are same.

In such a case you automatically use this technique. These cases of equations are called solving elimination equations. For, example, if x + y = 5 and x – y = 1, you eliminate ‘y’ by just adding both the equations to get the solution as x = 3. Similarly you eliminate ‘x’ by subtracting the second equation from first and figure out y = 2. But all systems are not as simple as this. Hence let us take a broader look on solving by elimination method.

Suppose, you have a system of equations in which all the coefficients of the variables are different. Select a variable for elimination. Your skill and experience will tell you which will be the ideal variable to eliminate. Take the LCM of the coefficients of that variable and multiply each equation with the ‘missing’ factor of the LCM in each case.

Now the given set of equations is transformed in such a way that the new set has the same coefficient for the selected variable in each equation. Now do the subtraction or addition operations on the transformed equations, so that the selected variable is eliminated and you get the solution of the remaining variable. Subsequently, plug in that in any of the given equations to figure out the solution of the other variable.

Let us elaborate with an example. Let the system be 17x + 2y = 49 and 19x + 3y = 54. It is prudent to decide to eliminate ‘y’. Eliminating ‘x’ not an incorrect step. But that will lead to cumbersome working. The LCM of 2 and 3 is 6.

Multiplying the first by 3 and the second by 2 (the missing factors of 6), the equations are transformed to 51x + 6y = 147 and 38x + 6y = 108.. A subtraction operation gives you the result as 13x = 39, means x = 3. Now plugging in x = 3 back, you can figure out y = -1.

The method of solving equations by elimination is the basis for the concept of use of determinants for solution of system of equations.

let us make a simple illustration. Suppose a1x + b1y = c1 and a2x + b2y = c2 . By elimination method you can establish, x = (c1b2 – c2b1 )/(a1b2 – a2b1 ), which is the concept behind Cramer’s rule of solution by using determinants.

Sequence and series

Sequence Series is a very important part of mathematics. When we write numbers in an order then we call it sequence of numbers.

The list of numbers follows a particular trend in that. The numbers in a seq are called terms of the seq. We cannot flip the terms in a seq.

Their order cannot be changed. We do the counting of terms from left to right.
For example: - 1, 3, 5, 7, 9 is a seq of five terms where first term is 1 and third term is 5 and last term is 7.

If we look at this seq we see that this seq follow a certain trend and that is every next term is two more than the previous term. Now what is a series?

A series is when we add the terms of a seq that constitutes to form a series. There are different types of seq and ser that exists and they are: -

1. Arithmetic Sequences and Series– In this type of seq and ser, the next term can be determined by adding the common difference to the previous term. The general arithmetic seq and ser is given as a, a+d, a+2d, a+3d, a+4d… where ‘a’ is the first term and d is the common difference. For example: - 1, 4, 7, 10, 13, 16. Here we see that first term is 1 and the common difference is 3

2. Geometric Sequences and Series– In this type of seq and ser, the next term can be determined by multiplying the common multiple by the previous term. The general geometric seq and ser is given as a, ar^2, ar^3, ar^4… where ‘a’ is the first term and r is the common multiple. For example: - 2, 6, 18, 54. Here we see that the first term is 2 and the common multiple is 3 which can be determined by dividing any term by its previous term.

3. Infinite Sequences and Series– In this type of seq and ser, the last term is unknown to us. We do not have fixed number of terms in that as it is an infinite ser. For example: - 2, 6, 10, 14…..

Series and Sequences Formulas are used to determine the first term, last term, sum of terms, number of terms or the common difference or ratio.

Equivalent Decimals

Decimals are nothing but a whole numbers but broken into pieces, so example we have one whole block, and we cut it in 2 pieces, and we call it as Tenths. How to do equivalent decimals, so how do we write it ? We write it as 1 .2 tenth (Here decimal represents AND). But if we break that down even further into more ten pieces and take out and add 3 then we call these as Hundredths.  So this way we have once, tenths and hundredths. So this way we will write it as 1.13 (one and thirteenth, remember here the last piece that we saw. Here we will name the last 1 and 3 as thirteenth, the last digit even if it is zero we call it that). Now late us take one more piece of cube and break it down into ten equal parts that will be called as thousands. So we have ones, tens, hundredths and thousands. Let us say we have this number up in broken to two pieces .we will call it as 1.232 that is one and two thirty two thousands, so we read the decimal like this way.

Now moving up to equivalent decimals examples, what if we have only one block so we will call it as 0.1( here we will read it as zero and one tenth.) but let us say we want to turn it to hundredths. The whole one piece when broken into ten equal parts has the same value for 0.1 as well. However they are hundredth. so all we need to say hundredth we need to add a zero. Because when we read decimals we need to say the last digit. Thus it will be 0.10(we call it as zero and one tenth one hundredth.) so one tenth is equal to tenth hundredths.

How to find equivalent decimals? We can by taking the denominator and dividing it with the numerator of the same fraction. Now for example we have ½ as a fraction so, 2 have to get divided by 1 so we will add a zero and bring up a decimal point. This way we have 10 which will be divisible by 10. As 2 times 5 is equals to 10.  And 10 subtracted by 10 and remainder comes to zero. So we have got the solution as 0.5 is the equivalent decimal.
What is a Equivalent decimal are nothing but the decimals which have the equal proportions.