Algebraic Method of Solving a Pair of Linear Equation

Algebraic Method of Solving a Pair of Linear Equations:

Substitution Method : We shall explain the method of substitution by taking
some examples.

Example 1 :
Solve the following pair of equations by substitution method:
7x – 15y = 2 (1)
x + 2y = 3 (2)
Solution :
Step 1 : We pick either of the equations and write one variable in terms of the other.
Let us consider the Equation (2) :
x + 2y = 3
and write it as x = 3 – 2y (3)

Step 2 : Substitute the value of x in Equation (1). We get
7(3 – 2y) – 15y = 2
i.e., 21 – 14y – 15y = 2
i.e., – 29y = –19
Therefore, y =19/29
Step 3 : Substituting this value of y in Equation (3), we get
x = 3 – 2 (19/29) =49/29

Therefore, the solution is x =49/29 , Y =19/29

Verification : Substituting x =49/29 and Y=19/29

,you can verify that both the Equations
(1) and (2) are satisfied.

Step 1 : Find the value of one variable, say y in terms of the other variable, i.e., x from
either equation, whichever is convenient.
Step 2 : Substitute this value of y in the other equation, and reduce it to an equation in
one variable, i.e., in terms of x, which can be solved. Sometimes, as in Examples 9 and
10 below, you can get statements with no variable. If this statement is true, you can
conclude that the pair of linear equations has infinitely many solutions. If the statement
is false, then the pair of linear equations is inconsistent.
Step 3 : Substitute the value of x (or y) obtained in Step 2 in the equation used in
Step 1 to obtain the value of the other variable.

Quadratic Equations

Quadratic equations:

Quadratic Equations come up when we deal with
many real-life situations. For instance, suppose a
charity trust decides to build a prayer hall having
a carpet area of 300 square meters with its length
one meter more than twice its breadth. What
should be the length and breadth of the hall?
Suppose the breadth of the hall is x meters. Then,
its length should be (2x + 1) meters. We can depict
this information pictorially as shown in Fig below


Now, area of the hall = (2x + 1). x m2 = (2x2 + x) m2
So, 2x2 + x = 300 (Given)
Therefore, 2x2 + x – 300 = 0
So, the breadth of the hall should satisfy the equation 2x2 + x – 300 = 0 which is a
quadratic equation.

Euclid’s five postulates

Euclid’s five postulates

Postulate 1 : A straight line may be drawn from any one point to any other point.

Axiom 5.1 : Given two distinct points, there is a unique line that passes through

them.

How many lines passing through P also pass through Q (see Fig. 5.4)? Only one,

that is, the line PQ. How many lines passing through Q also pass through P? Only one,

that is, the line PQ. Thus, the statement above is self-evident, and so is taken as an

axiom.

Postulate 2 : A terminated line can be produced indefinitely.

Note that what we call a line segment now-a-days is what Euclid called a terminated

line. So, according to the present day terms, the second postulate says that a line

segment can be extended on either side to form a line (see Fig. 5.5).

Postulate 3 : A circle can be drawn with any centre and any radius.

Postulate 4 : All right angles are equal to one another.

Postulate 5 : If a straight line falling on two straight lines makes the interior

angles on the same side of it taken together less than two right angles, then the

two straight lines, if produced indefinitely, meet on that side on which the sum of

angles is less than two right angles.

find the slope for the graph of the equation

Before we know how to find the slope for the graph of the equation ,lets see what is slope.

Slope is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. It is also always the same thing as how many rises in one run.

Using calculus, one can calculate the slope of the tangent to a curve at a point.

The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Question:-

Draw the graph of 2x-3y=12. Use the graph to find the slope of line and area between the axes.

Answer:-

This Online help with math shows us ,how to draw the graph for the above problem.

by simplifying the equation we get.

y=0, x=6
y=-2, x=3
y=-4, x=0

We also use the slope formula to get the slope on the line .

A problem on pythogoras theorm

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

The theorem can be written as an equation:

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.



Question:-

Calculate the value of cos θ in the above triangle.

Solution:-

Use Pythagoras theorem to evaluate the length of PR.

PR=√122+202

       adjacent
Cosθ= ----------
       hypotenuse

   RQ
= ----
   PR

    12
 = -----
   √544


 = 0.514 is the Answer

For more help on this ,you can reply me.

How to find the factors for an equation

Topic:-Factors of an equation

what is factorization : the decomposition of an object a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into prime factors as 3 × 5, and the polynomial x² − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained. However, factors are not needed to divide evenly because they are still divisible by any number. For example, technically 3 and 8/3 are factors of 8. But, when factoring for tests teachers are looking for the even divisibility of the numbers.
Let's look an example on it.

Question:-

Solve x²+16x+69=6

Answer:-
x²+16x+69=6

x²+16x+69-6=0

x²+16x+63=0

For factoring it ,we find two factors of 63
which add up to 16,they are 9 and 7

x²+9x+7x+69=6

Pair these terms

x(x+9)+7(x+9)=0

(x+9)(x+7)=0

x+9=0 0r x+7=0

x= -9 or x= -7

So x= -9 or -7 is the answer.

For more help on this ,you can reply me.

A word problem on Distance

Distance is a numerical description of how far apart objects are. In everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").

If we have a map which shows the distance between two cities and they have given distance as well.

Let's see how to find the actual distance ...

Question:-

On a road map ,if 10 miles is equal to 1 cm.Them find the value for 40 miles and find the actual distance as well.



Answer:-

10 miles=1 cm

40 miles=?

Let's take it as 'x'

We take the ratio and equate them.

10       40
----- = -------
 1        x

      40
 x = ------
      10


 x =4
So
40 miles = 4 cm on map

Now,using the same ratio


Distance on map = 45 cm

  we know 1 cm = 10 miles

Actual Distance = 45*10

               = 450 miles

For more help on this ,you can reply me.