Linear Equations

What is Linear Equations:

“A set of two linear equations { a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0 and { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0 in the same two variables x and is said form a system of simultaneous linear equations”

An equation of the form ax+by=c, where a,b,c are real numbers is called a linear equations in two variables x and y.

            The graph of a linear equations ax+by=c is a straight line.

            The graph of an equation in x and y is the set of all points whose co-ordinates satisfy the equation.

            For example, the linear equations 3x+2y=11, which we may write y=\frac { 11-3x }{ 2 } .

            Required table for y=\frac { 11-3x }{ 2 }.

x1-13
y=\frac { 11-3x }{ 2 }\frac { 11-3 }{ 2 }=4\frac { 11+3 }{ 2 }=7\frac { 11-9 }{ 2 }=1
linear equation diagram

        Clearly the point A(2,1) and B(4,2) do not lie on the given line i.e., these are not the solutions of this line(equation).

Linear Function

     Where, m= slope of the line,

                  c= intercept on the y-axis.

How to Draw the Graph of a Linear Equations?

                     Steps: ax+by+c=0

  1. Make y as the subject of the formula i.e.,y= mx + c.
  2. Select atleast two value of x, such that x,y \epsilon I.
  3. Make a table for the ordered pairs(x,y).
  4. Plot these points on a graph paper selecting suitable scale.
  5. Joint these points to get the graph of the line ax + by + c =0

Graphical Method of Solution of a Pair of Linear Equations

                 METHOD

                      Step 1: Read the problem repeatedly to detect the unknown which is to be found.

                     Step 2: Represent the unknown by x and y etc.

                     Step 3: Use the conditions given in the problem to frame an equation in the unknown x and y.

                    Step 4: Prepare the proper tables for both the situations.

                    Step 5: Draw the graph of both the equations on the same scale of representation.

                   Step 6: Detect to co-ordinates of the point of intersection of the graph.

Conditions for Simultaneous Equations

                         Two simultaneous equations

                              { a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0 and { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0

                   (i) Consistent if \frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ b_{ 2 } }

                   (ii) Inconsistent if \frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ b_{ 2 } } \neq \frac { { c }_{ 1 } }{ { c }_{ 2 } }

                   (iii) Dependent: \frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ b_{ 2 } } =\frac { { c }_{ 1 } }{ { c }_{ 2 } }

Solution of Simultaneous Linear Equations in Two Variables (BY Algebraic Method)

                      Given a linear equations in two variables x and y, we can find x in terms of y and y in terms of x but none in terms of known constants. But given two linear equations in two variables, it may be possible to find the values of x and y in terms of known constants satisfying the given equations. These values of x and y are called the solutions of the given simultaneous equations. Different methods for solving simultaneous linear equations in two variables are:

                  (i) Elimination by substitution either x or y.

                 (ii) Elimination by equating the co-efficients.

                (iii) By cross multiplication.

General Solution and Conditions for solvability

                        General equation of a linear equations in two variables is

{ a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0

                   We can also assume the general equation of a linear equation in two variables as

{ a }_{ 1 }x+{ b }_{ 1 }y={ c }_{ 1 }=

In the first equation,{ c }_{ 1 } is negative of { c }_{ 1 } in the second equation. Thus, the general system of two simultaneous linear equations in two variables is as

                                          { a }_{ 1 }x+{ b }_{ 1 }y={ c }_{ 1 }    …(1)

and                                   { a }_{ 2 }x+{ b }_{ 2 }y={ c }_{ 2 }    …(2)

            Multiplying eq.(1) by { b }_{ 2 } from eq.(2) by { b }_{ 1 }, we get

                                   { a }_{ 2 }{ b }_{ 2 }x+{ b }_{ 1 }{ b }_{ 2 }y={ b }_{ 2 }{ c }_{ 1 }    ….(3)

  And                        

{ a }_{ 2 }{ b }_{ 1 }x+{ b }_{ 1 }{ b }_{ 2 }y={ b }_{ 1 }{ c }_{ 2 }     ….(4)

            Subtracting eq. (4) form eq. (3), we get

{ a }_{ 2 }{ b }_{ 2 }x-{ a }_{ 2 }{ b }_{ 1 }x={ b }_{ 2 }{ c }_{ 1 }-{ b }_{ 1 }{ c }_{ 2 }

             Again, multiplying eq. (1) by { a }_{ 2 } and eq. (2) by{ a }_{ 1 }, we get

                           { a }_{ 1 }{ a }_{ 2 }x+{ b }_{ 1 }{ a }_{ 2 }y={ c }_{ 1 }{ a }_{ 2 }    ….(5)

And                   { a }_{ 1 }{ a }_{ 2 }x+{ b }_{ 2 }{ a }_{ 1 }y={ c }_{ 2 }{ a }_{ 1 }   ….(6)

        Subtracting eq. (5) from eq. (6) we get

{ a }_{ 1 }{ b }_{ 2 }y-{ a }_{ 2 }{ b }_{ 1 }y={ a }_{ 1 }{ c }_{ 2 }-{ a }_{ 2 }{ c }_{ 1 }

       Therefore, the solution of the given general system of equations is

x=\frac { { c }_{ 1 }{ b }_{ 2 }-{ c }_{ 2 }{ b }_{ 1 } }{ { a }_{ 1 }{ b }_{ 2 }-{ a }_{ 2 }{ b }_{ 1 } }, y=\frac { { a }_{ 1 }{ c }_{ 2 }-{ a }_{ 2 }{ c }_{ 1 } }{ { a }_{ 1 }{ b }_{ 2 }-{ a }_{ 2 }{ b }_{ 1 } }

Special Types of Problems

                  Suppose two linear equations in x and y are so given that the coefficients of x and y in one equation are interchanged in the other. In such also cases, we add and subtract the given equations to obtain them in the form

                      X + y =a

                      X-y =b 

IMPORTANT POINTS TO REMEMBER

  1. System of Simultaneous Linear Equations: A set of two linear equations in the same two variables is said to form a system of simultaneous linear equations.

         For example:

             (i) A system of equations: 2x + 3y =7 and x + y =1 is simultaneous since x=3, y=2 satisfy both equations.

            (ii) A system of equations: x + 2y =7 and x + 2y =9 is not simultaneous since no same values of x and y satisfy both the equations.

           (iii) A system of equations: x – y =6 and 2x – 2y =12 is also not simultaneous since second equation can easily be obtained from 1st by simply multiplying by 2. That is both are the same.

2. (i) Linear Equations in Two Variables: An equation which contains only power one of the variable(s) is called a linear equation.

A general equation in two variables x and y is written either in the form

                           Ax + by + c =0 or ax + by =d

(ii) Root: Any pair of values satisfying above equation is called the root or the solution of given equation.

(iii) The graph of ax + by + c =0 is always a straight line. Every point on the line gives a solution of the equation.

3. Techniques for Solving Two Simultaneous Equations (Algebric Method): Two simultaneous linear equations in two variables can be solved by three methods:

  1. Method of Substitution:

(i) Find the value of one of the variables in terms of other from any one of the given equation.

(ii) Substitute the value of the variable so obtained in the other equation.

(iii) Solve the equation thus obtained and find the value of one of the variables.

(iv) Substitute the value of the variable so obtained in any one of the given equations and find the value of the other variable.

  1.  Method of Elimination:

(i) Multiply the equations so as to make the coefficients of the variable to be eliminated equal.

(ii) Subtract/add the equations if the terms having the same coefficient are the same/opposite signs.

(iii) Solve the equation, find the value of one variable.

(iv) Substitute the value found in any one of the equations and find the value of the other variable.

4. If the equations are of the type:  { a }_{ 1 }x+{ b }_{ 1 }y=0;\quad { a }_{ 2 }x+{ b }_{ 2 }y=0

    Then (a) if \frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ { b }_{ 2 } } , the only solution is x = y =0

            (b) if \frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } }, the system has infinitely many solutions.

5. When a system includes equations with fractions as its coefficient of the variables, we take the L.C.M. of the denominators, and then multiply each term of the equation by its L.C.M. Thus we reduce the equations into its standard from. Thereafter, we solve the system of equations by any of the methods given above

6. Elimination by equating the coefficients.

   (i) Multiply the equations so as to make the coefficients of the variable to be eliminated equal.

  (ii) Subtract or add the equations if the terms having the same coefficient are of same or opposite sign.

 (iii) Solve the equations, find the value of one variable.

 (iv) Substitute the value found in any one of the equations and find the value of the other variable.

 (v) Check the solution in both given equations.  

Final Wordings:

You have made it to the end of our introduction to the Linear Equations. We have decided to write all the Examples and Sample Questions which comes under the Number System chapter. You can find articles on both topic under the Example And Question Bank tab.

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Thanks for visiting our site dedicated to the Linear Equations.

Further Information:

Source-: Ts aggarwal & R.L Arora References


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