Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables

1. Geometric [Graphical] Solution

   • Single and Unique Solution
     • Intersecting lines
       • Consistent
       • If the Pair of Linear Equations are ax+by+c= 0 and Ax+By+C=0 then a/A ≠ b/B ≠ c/C
   • No solutions at all
     • Parallel Lines
       • Inconsistent
       • If the Pair of Linear Equations are ax+by+c= 0 and Ax+By+C=0 then a/A = b/B ≠ c/C
   • Infinitely Many Solutions
     • Identical or Overlapping Lines
       • Consistent
       • If the Pair of Linear Equations are ax+by+c= 0 and Ax+By+C=0 then a/A = b/B = c/C

2. Algebraic Solution

   • Substitution Method
     • We Substitute the Value of one Variable by Expressing it in the Term of the Other Variable to Solve the Pair of Linear equations
   • Elimination Method
     • We Eliminate one Variable First to Get a Linear Equation in one Variable and Then Solving it we Find the other Variable
   • Cross Multiplication Method
     • If the Pair of Linear Equations are ax+by+c= 0 and Ax+By+C=0 then x/(b*-B*c)= y/(c*A-C*a)= 1/(a*B-A*b)

3. Equations Reducible to a Pair of Linear Equations

   • in Pair of Equations with Variables in the Denominator we change the Fraction To Another Variable
     • Eg. Let the Pair of Equations be 2/x+3/y= 13 and 5/x-4/y= -2
       • We take (1/x) as 'p' and (1/y) as 'q'
         • Then the Equations are 2p+3q=13 and 5p-4q=-2
           • After solving for 'p' and 'q' we solve for (1/x)=p and (1/y)=q