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Linear Equation Two Variable2

Linear Equation Two Variable2 is the continuation of previous page .


4. Check whether given pair of line is consistent or not 5x - 1 = 2y, y = \(\frac{-1}{2} + \frac{5}{2}x.

Solution:-

                The lines \(a_1x + b_1y + c_1 = 0\),\(a_2x + b_2y + c_2 = 0\) are consistent if \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) .

                                            5x - 2y - 1 = 0
                                            \(\frac{5}{2}x\) - y - \(\frac{1}{2}\) = 0

                                            \(\frac{5}{\frac{5}{2}} = \frac{-2}{-1} = \frac{-1}{\frac{1}{2}}\)

                                             2 = 2 = 2       \(\therefore\) The given condition is true

\(\therefore \) The lines are consistent.


5. Write any one equation of the line which is parallel to √2x - √3y = 5.

Solution:-

       The lines \(a_1x + b_1y + c_1 = 0\),\(a_2x + b_2y + c_2 = 0\) are parallel if      \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).

    Let us think of a line

                        2√2x - 2√3y = 15

    To verify this ,

                                     \(\frac{√2}{2√2} = \frac{-√3}{-2√3} \neq \frac{5}{15}\)

                        \(\frac{1}{2} = \frac{1}{2} \neq {1}{3}\)  

Therefore it is true .

Linear Equation Two Variable2

6.Find the point of intersection of line -3x + 7y = 3 with x-axis.

Solution :-

            To get the point of intersection with x-axis we need to get y value '0' that is the ordered pair must be in the form of (a, 0) .

            So to find the value of 'a' we must plug the value of y that is y= 0,

                                          -3x + 7y = 3

            Put y = 0 and x = a, we get,

                                          -3(a) + 7(0) = 3

                                          -3a = 3

                                          a = -1 

Therefore the point of intersection of line -3x + 7y = 3 with x-axis is (-1 , 0).


What We Learn :

  • A pair of linear equations in two variable, which has a solution is called consistent pair of linear equations.
  • A pair of linear equations in two variable, which has infinitely many solutions are called dependent pair of linear equation..
  • A pair of linear equations in two variable which has no solution is called inconsistent pair of linear equation.

    NOTE :-  Every dependent pair of linear equations is a consistent pair of linear equation, but every consistent pair of linear equations may not be a dependent pair of linear equations. 

  • If the ratios of coefficients of 'x' terms is equal to the coefficients of 'y' terms and not equal to the rations of constant terms, then the pair of linear equations are parallel.
  • To get the intersection point of a given equation with x-axis we need to plug the value of y as 0, (i.e., the point on x-axis is in the form of (a,0)).


These are the points we learn in this page of topic Linear Equation Two Variable2.

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