1. Linear equations in two variables have the form ax + by = c, where a, b, and c are real numbers and a and b are not both zero. 2. The graph of a linear equation in two variables is a straight line. 3. There are three types of systems of linear equations: intersecting lines, coincident lines, and parallel lines. Intersecting lines have a unique solution, coincident lines have infinitely many solutions, and parallel lines have no solution.

1. Linear Equations in Two Variables Linear Equations MADE BY : SHASHI PRKASH Xth–C 35

2. y (0,4) (6,0) x 2 -2 Equations of the form ax + by = c are called linear equations in two variables.Where A, B, and C are real numbers and A and B are not bothzero. This is the graph of the equation 2x + 3y = 12. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept. The graph of any linear equation in two variables is a straight line Linear Equations

3. INTERSECTING LINE 𝒂𝟏𝒃𝟐≠𝒃𝟏𝒃𝟐 are called INTERSECTING LINES Equations of the form These type of equations have Exactly one solution(unique) These are consistent lines Example: x-2y=0 3x+4y=20 𝑎1𝑎2= 13 𝑏1𝑏2= −24 𝑎1𝑎2≠𝑏1𝑏2

4. COINCIDENT LINES 𝒂𝟏𝒃𝟐=𝒃𝟏 𝒃𝟐=𝒄𝟏𝒄𝟐 are called COINCIDENT LINES Equations of the form These type of equations have Infinity many no. of solution. These are consistent lines Example: 2x+3y=9 4x+6y=18 𝑐1𝑐2=12 𝑎1𝑎2= 12 𝑏1𝑏2=12 𝑎1𝑎2=𝑏1𝑏2=𝑐1𝑐2

5. PARALLEL LINES 𝒂𝟏𝒃𝟐=𝒃𝟏 𝒃𝟐≠𝒄𝟏𝒄𝟐 are called PARALLEL LINES Equations of the form These type of equations have no solution. These are inconsistent lines Example: x+2y=4 2x+4y=12 𝑐1𝑐2=13 𝑎1𝑎2= 12 𝑏1𝑏2=12 𝑎1𝑎2=𝑏1𝑏2≠𝑐1𝑐2

6. X + y=10 X - y=4 EXAMPLE : Graph

7. Example: x + y=5 2x+2y=10 Graph

8. Example: 2x + y=160 2x+y=150 Graph

9. Substitution Method EXAMPLE : x + y=14 x – y=14 Solution : x + y=14-------------------------1 x – y=14-------------------------2 From equation 1 x =14-y x =14-5 X=9 Put the value of x in equation 2 14-y-y=4 14-2y=4 14-4=2y 10=2y Y=5

10. Elimination Method EXAMPLE : x + y=25 x + 2y=40 x + y=25-------------------------1 Solution : x + 2y=40-------------------------2 Subtract equation 2 from 1 x + y=25 x + 15=25 X=25-15 X=10 x + 2y=40 - - - -y=-15 -y=-15 Y=15

11. Cross-Multiplication Method The general form of cross- multiplication method is: 𝒙𝒃𝟏𝒄𝟐−𝒃𝟐𝒄𝟏=𝒚𝒄𝟏𝒂𝟐−𝒄𝟐𝒂𝟏=𝟏𝒂𝟏𝒃𝟐−𝒂𝟐𝒃𝟏