Solving by Substitution Method

Algebra 1 Help : In the algebra there are several system of equations that have the several variable so values of these variables are solved by Substitution method. When there are several equations in the system then these are solved using some predefined steps of substitution method to solve algebra linear equations that are as follows :
( a ) There are 1 variable for one equation as x ,y,z means if there are three variable given in an equation then there are also three equations to find the value of these variables .
( b ) For solving the missing variable substitute the expression into the other equation .
( c ) When find the value of a variable that find into the step (b) then put it into the first equation and find the answer .
( d ) Cross check the solution .
The above steps are using to find the values of the variable in the given example :
-p +q = 1
2 p+q = -2
step no ( a ) : Find one equation for one variable there are two variable p and q
-p+q = 1
-p+p+q = 1+p (add the +p for both side of equation to find the one variable )
q = 1+p or q = p+1 .
step no ( b ) : For solving the missing variable substitute the expression into the other equation as
2 p+q = -2
put the value of q by the step no ( a ) into this equation
2 p+(1+p) = -2
2 p+p+1 = -2
3 p+1 = -2
Solve the value of p by subtracting the 1 from both side of equation
3 p+1–1 = -2–1
3p = -3
Divide the both side of equation by 3
3p/3 = -3/3
p = -1
Step no ( c ) : Find the value of q by putting the value of p into one of the equation
q = p+1
q = -1+1
q = 0
step no ( d ) : Cross check the answer
put both the values into each equation
-p+q = 1 2p+q = -2
-(-1)+0 = 1 2(-1)+0 = -2
1 = 1 -2 = -2


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Solving by Elimination Method

What is a linear equation? Equation is a collection of terms, which contains the combination of variables and numbers to represent the certain values. Sometimes linear equation can be described as a statement of variables to represent the particular values in the mathematics. As we know that mathematics provides lots of tools like method, formulas, theorem and rules. We will discuss about Solving by Elimination Method for the equations which having more than two unknown variables. In the general aspect we include the problem, which is given below:
x + y = 13        (1)
x – y = -5         (2)
The above are the equation, in which we can see that there are two equations containing two unknown variables x and y. If we determine the values of above two unknown variables x and y, which are consider as a true value for equation (1) and (2) then we can say that equation as a simultaneous  equations. To solve the simultaneous equation, we generally makes the value of one unknown variable same in both equation by adding or subtracting the variable to form new equation, which contain only one variable. After that obtain the value of one variable is easy to find. This whole process is known as Elimination Method.
 Here we show you how to solve the equation by example:
Example: Solve the equation by elimination method of following equations?
                         4y + 3x = 9
                         3y - 3x = 5
Solution: first we give label to both given equations:
                         4y + 3x = 9         (1)
                         3y - 3x = 5       (2)
Now we add the equation (1) and (2) in the same manner:
            4y + 3x + 3y - 3x = 9 + 5         (here +3y and -3y are cancelled to each other)
                         7y  = 14
                            y = 14/ 7
                            y = 2

In upcoming posts we will discuss about Solving by Substitution Method and Right Triangle. Visit our website for information on higher secondary education Karnataka

Finding Solution of Simultaneous Equations by Graphing

Hello students in this session we are going to talk about Finding Solution Of Simultaneous Equations By Graphing. But before discussing it you should know about linear equation, it is an algebraic expression that contains either a static term or the outcome of a static and one variable. Meaning of Simultaneous is ‘occurring at the same time’ that means we have to find Solution of Equations when they are occurring at the same time. (get detail)
The meaning of this is a pair of linear equation in two variables is said to form a system of simultaneous linear equation like
x + 2y = 3, 2x – y = 5
We can find Solution of Simultaneous Equations by Graphing with the following methods and cases they are:-
Solving Linear Equations
Method :- let the given system of linear equation be
a₁ x + b₁ y + c₁ = 0
a₂ x + b₂ y + c₂ = 0 (or try linear equations calculator)
On the same graph paper, we draw the graph of each one of the given linear equation. Each such graph is always a straight line. Let’s suppose we have two lines L₁ and L₂ that can be represented in a graph. Then many cases arise, some of them are :-
Case 1 :- When the lines L₁ and L₂ intersect at a point.
Case 2 :- When the lines L₁ and L₂ are coincident. It means they have infinitely many common points.
Case 3 :- When the lines L₁ and L₂ are parallel. It means they do not have a common point and so the system has no solution that is non-consistent. If they have at least one solution then the system is consistent.
We can make a algorithm for the above mentioned method and cases by marking them step 1, 2 to 5 so that we can easily solve Simultaneous Equations by Graphing .


In upcoming posts we will discuss about Solving by Elimination Method and Isosceles Triangle. Visit our website for information on ICSE syllabus for business studies

Substitution Method to Solving Simultaneous Equations

If two linear equations are solved at the same time then these equations are known as simultaneous. We understand the simultaneous equations by the help of some examples as
p + q = 5 and p – q = 1 are described as the simultaneous equations (more detail here).
Simultaneous equations can be solved exactly with the help of either substitution method or elimination method. Here we will use Substitution Method to Solving Simultaneous Equations. (or try linear equation calculator)
We take an example of substitution Method to Solving Simultaneous Equations as follows:
p + q = 3
2 p + 3 q = 8 (you can also try linear equation solver)
Both the equations have the sane variables p and q and both have the same solutions, so these are simultaneous equations p = 1 , q = 2 .
Substituting p = 1 and q = 2 in both the equations:
1 + 2 = 3 and 2 * 1 + 3 * 2 = 8
3 = 3 and 2 + 6 = 8 that is 8 = 8
Thus the solutions of variables p = 1 and q = 2 is correct.
For solving simultaneous Equations by the substitution method we have to follow some steps as :
step 1 : From one side of equations pick the one variable ( p )
p + q = 3
Isolate p : p = 3 – q
Step 2 : In other equation isolate the other variable :
2 p + 3 q = 8
Substitute 3 – q in place of p
2 ( 3 – q ) + 3 q = 8
This above equation has only single variable so it can solve easily .
Step 3 : For another variable q solve this equation :
2 ( 3 – q ) + 3 q = 8
Brackets are expanded as :
6 – 2 q + 3 q = 8
6 + q = 8
q = 8 – 6 = 2
q = 2
Substitute the q = 2 in equation for getting p
p = 3 – q
p = 3 – 2
Then p = 1


In upcoming posts we will discuss about Finding Solution of Simultaneous Equations by Graphing and Equilateral Triangle. Visit our website for information on CBSE board home science syllabus for class 11

Solving Linear Equations Examples

Hi friends, I am going to discuss about how to graph linear equations?
Linear equation is a pair of equation in two variables. There are two types of linear equations. First one is simultaneous linear equation of two variables and second one is graphical representation of linear equation. In this session, we will be discussing solving linear equations with two variables and solving simple problems from different areas.
In general form a linear equation in two variables x and y is
a₁x+b₁y+c₁=0
a₂x+b₂y+c₂=0
where a₁, b₁, c₁  and a₂, b₂, c₂ are all real numbers and x, y are variables. This is known as the algebraic representation of linear equation in two variables.


We take some examples of linear equation.
Example 1:- Show that x=2, y=1 is a solution of the simultaneous linear equations.
3x-2y=4
2x+y=5
solution :- the given system of equation is
3x-2y=4
2x+y=5
putting the x=2 and y=1 in equation (1) we have
L.H.S= 3*2-2*1=4= R.H.S
putting the x=2 and y=1 in equation (11) we have
L.H.S=2*2+1*1=5=R.H.S
thus, x=2 and y=1 and this satisfies both the equations of the given system.
Hence, x=2 and y=1 is a solution of the given system.
Now, we discuss about linear equation with fraction such that express by x/y.
We take some example to solving linear equations with fractions
Example 1:- solution equation (x+1)/3 =(2x+1)/5 solving by linear equation with fraction.
Solution :- to solve the equation step1:- (x+1)/3 =(2x+1)/5
step 2:- (x+1)/3 =(2x+1)/5 (we can start with left side and we can use cross multiply it means to one multiply numerator of one fraction by denominator of another fraction).
Step 3 :- 3(2x+1) = 5(x+1)
step 4:- 6x+3 =5x+5
step 5:- 6x-5x = 5-3
step 6:- x = 2 (this is answer for x)
now, we can check this solution putting the x = 2 in equation (1) we have
step 1:- L.H.S = (2+1)/3 =3/3 = 1
now, again putting the x = 2 in equation (2) we have
step :- R.H.S = (2*2+1)/5 =5/5 =1
so, L.H.S =R.H.S

In upcoming posts we will discuss about Substitution Method to Solving Simultaneous Equations and Congruent Triangles. Visit our website for information on CBSE board fashion studies syllabus for class 11

Linear Equations Word Problems

Students, we have already learned about  Linear Equations in earlier sessions. Linear Equation problems are used to solve daily life problems. It is applicable to find the values of the unknown values. Here we are going to look at Linear equations Word Problems, which will help us to get the solution to the unknown values. This solution can be attained by forming proper equation by reading the statements given in the word problems. It is done by understanding the meaning of the words used to represent the mathematical operators in the given statements. Let us understand it more  clearly through some example:
Solve Equation:
Problem 1 : Five added to twice a number gives eleven. Find the number?
  Sol: Let us assume that the number is x.
Now As per the statement, twice a number represents two times a number means 2 * x
 And 5 added to  twice a number will be expressed as ( 2 * x ) + 5
 So finally the equation formed is as follows
        ( 2 * x ) + 5 = 11
 Now we proceed to solve the equation to find the value of x
      Subtracting 5 from both the sides we get
       ( 2 * x ) + 5 -5 = 11 -5
      ( 2 * x )  = 6
     or  x = 6 / 2
    or x = 3 Ans.
Problem 2: If 3 is subtracted from four times a number, gives 13. Find the number.
  Sol:  Let the number is y
        Four times a number is represented by 4 * y
        Further if 3 is subtracted from it , we get 4 * y  - 3
       Thus the equation takes the form:  (4*y ) - 3 = 13
      Now to solve the equation, we add 3 on both the sides and get:
         (4*y ) - 3 + 3 = 13 + 3
        (4*y )  = 16
        or y = 16 / 4
      or y = 4 Ans.

In upcoming posts we will discuss about Solving Linear Equations Examples and Base and Altitude. Visit our website for information on CBSE board papers of political science

Elimination Method to Solve Simultaneous Equations

Today we are going to learn about Simultaneous solving equations. In case we have two equations with two variables to be solved , we call them Simultaneous Equations. We have different  Method to  Solve Simultaneous Equations, among which most common are LPP and Elimination Method to Solve Simultaneous Equations. (Also improve your skills in graphing linear equations)

Here we are going to learn about Elimination Method to Solve Simultaneous Equations.
In this method, let us say we have two variables x and y in the given two equations. We first find the value of x in form of y and put that value of x in the second equation.
By this method, we eliminate one variable from the equation and the equation is left with only one variable. So we simply find the solution of the equation in one variable .
In this way the value of one variable is solved. Putting the value of one variable in the first equation and we get the value of another variable also. Get more detail here.
Let us learn it through an example:
Solve the given pair of equations: x + y = 10 ------ (1)
                                               and 2x + y = 14 ----- (2)
Sol : from First equation: x + y = 10
     We can  get      y = 10 - x  ----- (3)
   From the above  we put the value of y from  (3)  in (2)
  This is called elimination of y from (2) equation.
we get , 2x  + ( 10 - x ) = 14
  or, 2x +10 - x = 14
  or,  2x - x = 14 - 10
 or, x = 4 is the value of x
Now putting this value in (3)  we get
  or, y = 10 - 4
 or  y = 6 Ans.

In this way two values of x and y can be solved by eliminating one variable from the equation.

In upcoming posts we will discuss about Linear Equations Word Problems and How to find Median. Visit our website for information on CBSE board physics syllabus

Math Blog on How to Solve Linear Equations

Hi students! In this learn algebra online session we are going to learn about ways to solve linear equations. Linear equations that are given as problems can be of single variable or of double variables.
For solving Linear equation of single variable, we just add, subtract, multiply or divide with the same number in such a way that it ends up with only variable on one side and with the solution on the other side.  This method of solving the linear equation is also called “solving linear equations by elimination”. Let us understand it more clearly with an example:

 Solve: 2* y + 6 = 10
 Now first we eliminate 6 from left hand side. For this, we subtract 6 from both the sides.
We get:
 2*y + 6 - 6 = 10 - 6
Or 2* y = 4
Now in next step, we will eliminate 2 from left hand side, so that we are left with y only.
For this we divide both sides of the equation by 2
=> 2*y / 2 = 4 / 2
Or y = 2 is the solution.
Let us take another example:
5*x -10 = 2*x + 20
In first step we will eliminate -10 from left side of the equation.
This will be done by adding 10 on both the sides:
5*x -10 + 10 = 2*x + 20  + 10
 or 5* x = 2 * x + 30
Now in second step we will subtract 2x from both the sides. This will help to eliminate 2x form right side of the equation.
We get:
5* x - 2x = 2 * x + 30 - 2x
 Or 3x = 30
 Now we observe that only 3 is to be removed from LHS, so we divide both sides by 3.
=> 3x/3 = 30 / 3
or x = 10 Ans.

In upcoming posts we will discuss about Elimination Method to Solve Simultaneous Equations and Acute Triangle. Visit our website for information on CBSE computer science syllabus

Linear Algebra

Hello friends, today we are going to learn algebra that is used to study about the set of linear equations.

Linear equation in terms of n variables is shown as a₁x₁+a₂x₂+......+a_nx_n=b. Where a₁,a ₂ up to a_n and b are the real numbers.

Example : a₁=2, a₂=3 and b=6 then equation will be 2x+3y=6 that describes the line passing through points (2,0) (0,3).

Matrices and determinants are most useful thing in the linear algebra. you can get more help on algebra here

Solutions of an equation with the help of matrices:

The equation is expressed as Ax=B. By solving the equation we get x in terms of inverse matrix as x=A^-1.B. Here A^-1 is showing the inverse matrix of A.

A vector space is a set V with two binary operations over a field that satisfies the axioms of the algebraic equation. In the vector space V and the operation of addition, multiplication etc follows some axioms as :-

Associativity of addition : a+(b+c)=(a+b)+c

Commutativity of addition : a+b=b+a

Identity element of addition : Є V as V+0=V

Inverse elements of addition as For every v ∈ V, there exists an element −v ∈ V, called the inverse element v such as v+(-v)=0

Here also some rules related with the multiplication :

Distributativity of scaler multiplication of a vector addition : a(b+c)=ab+ac

Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv

Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicativity identity.

In upcoming posts we will discuss about Math Blog on How to Solve Linear Equations and Base angle. Visit our website for information on secondary school education Andhra Pradesh

Application of Linear Algebra

Linear algebra on the whole is studying about the set of linear equations. These problems are specified in mathematics, physics, and also in engineering.
Linear algebra deals with vector space. A vector space is a set K with two binary operations over a field that satisfies the axioms of the algebraic equation. In the vector space K and the operation of addition, multiplication etc follows some axioms as:-
Associatively of addition, a+(b+c)=(a+b)+c
Matrix and determinant are most useful thing in the linear algebra. The main problem of linear algebra is solution of the matrix. The linear algebra solution is expressed as Ax=B. By solving the equation we get x in terms of inverse matrix as x=A-1.B. Here A-1 is showing the inverse matrix of A. Linear equation in terms of n variables is shown as a1x1+a2x2+......+anxn=b, where a1,a 2 up to an and b are the real numbers.
We will understand it by an example or you can use online tools like linear equation calculator.
If a1=2, a2=3 and b=6 then equation will be 2x+3y=6 that describes the line passing through points (2, 0) (0,3).

Applications of linear algebra are following:-
1.      Linear algebra is used in different areas like:-abstract thinking, games, cryptography, economics, image compression etc.
2.      In cryptography, people concerned with private communication use the encryption and decryption method with the secret key. Fundamentally it is used for security purpose.
3.      Linear algebra is also used in image compression. It is used in compressing digital images and it takes less space when stored and transmitted.
4.      Linear programming is another approach to linear algebra problems. We will see in which way we are maximizing or minimizing a linear expression in any number of variables with some linear constraints. This technique is used in a wide range of problems in industry and science.

In upcoming posts we will discuss about Linear Algebra and Vertex angle. Visit our website for information on board of intermediate education Andhra Pradesh