Simultaneous equations, Mathematics

Assignment Help:

Before we look at simultaneous equations let us brush up some of the fundamentals. First, we define what is meant by an equation. It is a statement which indicates that two algebraic expressions are equal. For instance, let 3x - 4 be an expression and 5x - 10 be another expression. If these two expressions are related to each other by an equality sign in the fashion shown below we call it as an equation.

                   3x - 4 = 5x - 10                                         .......... (1)

The side on which we have the expression 3x - 4 is referred to as Left Hand Side (LHS) and the one which has 5x - 10 as the Right Hand Side (RHS). If we substitute x = 3 in the above equation we find that both sides of the equation gives us 5. Now we substitute some other value say x = 2. We find that the LHS gives us 2 whereas the RHS gives us 0. Looking at these two cases we conclude that only when x = 3, the equation holds and for other values of x it does not. But consider an equation which is shown below.

                   3x + 2 + 2x - 5 = 5x - 3                            .......... (2)

The LHS and the RHS of this equation gives us the same values for any value of x. In other words, this equation holds for any value of x. Equations like these are called identities and the one we have seen before are referred to as equation of condition or more simply as an equation.

Above we have seen that only when we have substituted x = 3 in equ.(1), it holds true. That is, the value of x = 3 is said to be satisfying the equation. Since we are expected to find the value of x for which the equation holds true, the quantity x is known as the unknown quantity. The value of x found after solving the equation is called the solution or the root of the equation.

While solving equations, we have to remember these points.

1. If we are to add or subtract any quantity from/to one side of the equation, we should do so for the other side also. We look at this by taking an example.

For instance, we are required to solve the equation

                            x + 3 = 15

That is, on the LHS we ought to have only x. Can we subtract 3 from the LHS so that +3 and -3 cancel each other leaving behind only x? We can. But as stated above this operation should be done on both sides of the equation. That is, we will have

  x + 3 - 3 = 15 - 3

  x + 0      = 12

  x           = 12

If we do not perform this operation on both the sides, the balance which exists between the sides gets disturbed as a result the equality sign loses its relevance and it no longer has any meaning.

We take another example and check the same for addition. We have an equation x - 3 = 12, for which we have to obtain a solution.

x - 3 + 3 = 12 + 3

x = 15

As we are aware of this, while solving equations we directly transpose the quantity to the other side of the equation with its sign changed. Here we introduced a new word "Transpose". What is meant by Transposing? Bringing any term from one side of the equation to the other side is called transposing.

2. If we are to multiply or divide a particular element or the whole expression on one side of the equation, then we should do the same on the other side of the equation also. Let us take an example and understand this. We have to find the solution for the equation 3x + 5 = 20. We begin by subtracting 5 from both the sides. That will be

3x + 5 - 5 =20 - 5

3x + 0 =15

3x =15

Since only x ought to be there on the LHS (i.e. solving for x), we divide the LHS by 3 and do a similar operation on the RHS also. We have

3x/3 = 15/3

1.x =5

x =5

Therefore, x = 5 is the solution of the given equation.

Suppose we are given an equation like

(x-4)/3= 6

and asked to solve, how should we proceed?

We begin by multiplying both the sides of the equation by 3. We have

(x-4)/3 x 3 = 6 x 3

x - 4 =18

x - 4 + 4 =18 + 4

x =22


Related Discussions:- Simultaneous equations

Equations in linear algebra and matrices, Equations in linear algebra and m...

Equations in linear algebra and matrices What is Equations in linear algebra and matrices?

Patrice has worked a certain how many hours has she worked, Patrice has wor...

Patrice has worked a certain amount of hours so far this week. Tomorrow she will work four more hours to finish out the week along with a total of 10 hours. How many hours has she

What was the total cost of her order, Leslie ordered a slice of pizza for $...

Leslie ordered a slice of pizza for $1.95, a salad for $2.25, and a soda for $1.05. What was the total cost of her order? The cost of every item must be added together; $1.95 +

Infinity, Are there more rational numbers than integers?#

Are there more rational numbers than integers?#

Solve the right triangle, 1. Solve the right triangle. B = 135     c = 3...

1. Solve the right triangle. B = 135     c = 3.72 A  ≈ ____°    (round to the nearest tenth as needed) 2.  Solve the right triangle, where  a =4 and b =10 The length of

Divergence test - sequences and series, Divergence Test Once again...

Divergence Test Once again, do NOT misuse this test.  This test only says that a series is definite to diverge if the series terms do not go to zero in the limit.  If the

Find the distance between these two cities, Memphis, Tennessee, and New Orl...

Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has latitude 35°N and New Orleans has latitude 30°N. Find the distance between these

Conscious consumer, a group of 3o students is planning a thanksgiving party...

a group of 3o students is planning a thanksgiving party items needed hats @ $2.50 each.noise makers@$4.00 per pack of 5.Ballons @$5.00 per pack of 10.how many packs of noisemakers

The mode -measures of central tendency, The mode - It is one of the me...

The mode - It is one of the measures of central tendency. The mode is defined as a value in a frequency distribution that has the highest frequency. Occasionally a single valu

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd