Linear equations in two variables

1. Algebraic method of solving pair of equations in two variables

2. Elimination method- Step 1- Make the coefficients of one of the variable numerically equal Step 2- add or subtract so that one of the variable is eliminated Step 3- Solve the equation to get value of one variable and use it to find value of the other variable For example- 2x + 3y = 7 4x + 3y = 11 On subtracting eq2 from eq 1 we get-> (-2x)=(-4y) => x= 2 Now using x=2 in eq 1- 2×2 +3y =7 3y=3 y= 1 This is how elimination method is used to solve pair of linear equations in two variables

3. Graphical method of solving these equations

3.1. Step 1- get three or two solutions of each equation Step 2- plot these lines on a graph Observation- if lines are parallel then there is no common solution if the lines are intersecting then there is one solution if lines are coincident then there are infinitely many solutions

3.1.1. If a1/a2 = b1/b2 ≠ c1/c2 then the lines will be parallel and equations will have no solution

3.1.2. If a1/a2 ≠ b1/b2 then these lines will intersect and have a unique solution

3.1.3. If a1/a2 = b1/b2 =c1/c2 then the lines will be coincident and have infinitely many solutions

4. Equation that can be written in the form of ax+b=0 where a≠0

4.1. Two linear equations are represented in the form a1x + b1y + c1=0 a2x + b2y + c2=0

5. Substitution method- Step 1- find value of one variable in terms of other variable Step 2- substitute that value in other equation and solve Step 3- we get value of one of the variables and can be used to find the value of other variable For example- 2x-y=1 and 4x +3y= 27 From eq 1 we get y=2x-1 and we substitute this value in eq 2- 4x +3(2x-1)= 27 => 10x=30 => x= 3 Putting the value of x in y=2x-1 y= 2×3-1 y= 5 This is how equation can be solved with substitution method

6. Equations reducible to pair of linear equations

6.1. If the given pair of equations are not linear they can be reduced to linear form by making suitable substitutions then we reduce and solve For example- If the given equations-> 2/x + 3/y = 13 5/x - 4/y = -2 Here we will take 1/x=p and 1/y = q So the equations become-> 2p+3q =13 5p-4q = -2 After solving these equations either graphically or algebraically we get-> p=2 and q = 3 Now x=1/p => 1/2 y=1/q => 1/3 This is how we reduce the equations and solve them