Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE Main or Advanced/AIEEE, and anyone else who needs this Tutorial as a reference!After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well. Basics  what a straight line is:A line is an infinite geometrical figure. If you extend a line segment at both ends, you get a line. A line is always represented by two arrows at its ends, to indicate its infinite nature. Mathematically, a line can be represented by a linear equation, that is, an equation of degree one. The most general form of a straight line is y = mx + c. All the points (x,y) that lie on a line satisfy the equation of that line, and conversely, if a point (x,y) satisfies the equation of a line, it lies on that line.Plotting a line on the Cartesian Plane :To plot a line in the Cartesian plane, you need at least two points. Join the points with a straight line and extend it in both directions. The easiest way to get those points is to put x=0 and y=0 and get thecorresponding points as (0,b) and (a,0). The points A(a,0) and B(0,b) where the line intersects the x and y axis respectively are called intercept points. a and b are the lengths of intercepts. And OA and OB are called as X and Y intercepts respectively. NOTE: If a line is parallel to xaxis, that is represented by the equation of the type y=±k, then there is no Xintercept. Similarly, a line parallel to yaxis, that is represented by the equation of the type x=±k, then there is no Yintercept. k in the above equations represent the perpendicular distance from the axis to which line is parallel. Click on the images below to play around with 2 visualizations which demonstrate the line and its equation in these two forms : 1) Equation of a line given the two intercepts 2) Equation of a line in terms of its slope and YIntercept Slope of a Line:The tangent of the angle that a line makes with the positive xaxis in counter clockwise direction is defined to be the slope of the line. Slope is represented by the letter ‘m’, and indicates the steepness ofline in question. If (x1,y1) and (x2,y2) are two points on a line, then slope can also be found as m = (Y2Y1) / (X2X1) Line Parallel to the XAxis:y = k
Line Parallel to the YAxis:( Image to right : Screenshot of the linedrawing applet. )
General Equation of a Line:y = mx + c
Where m is the slope and c is the intercept on the YAxis Equation of a Line using the Intercept formIf it is given that a line intersects the X axis at (a,0) and the Y axis at (0,b) the equation of the line is : x/a + y/b = 1 ( a and b are the X and Y Intercepts respectively )
Examples:
What is the equation of the line joining (11,100) and (12,110) ?
Slope of the line = ( 110  100 ) / ( 12  11 ) = 10/1 = 10
Let us write the equation as : y = 10x + c
Since the line passes through (11,100) , substitute y=100 and x=11 in the above equation :
100 = 10 * 11 + c ==> c = 100  110 = 10
==> Equation of the line is : y = 10x  10
Finding the point of intersection of two linesWhat is the intersection point of the following lines :
1) 5x + 2y = 100
2) 3x  3y = 81
To get the intersection point, solve the two equations simultaneously.
Multiply (1) by 3 and (2) by 5 to get :
15x + 6y = 300
15x  15y = 405
Now subtract the lower equation from the upper equation : 21y = 105 ==> y = 5
Substituting y = 5 in equation (1) :
The angle θ between two lines having slope m1 and m2 is given by tan ^{1} m_{1}  tan ^{1} m_{2} Conditions for Two Lines to be Parallel or PerpendicularTwo lines with slopes m1 and m2 are: parallel if m1=m2 perpendicular if m1 . m2 = 1 Form of an equation of a line parallel or perpendicular to a given lineEquation of line parallel to ax+by+c=0 is ax+by+k=0. (k is any constant) Equation of line perpendicular to ax+by+c=0 is bxay+k=0. Condition for Collinarity of Three PointsThree points P,Q,R are collinear if slope(PQ)=slope(QR) [This is the best collinearity test] Identifying Slope, XIntercept, YIntercept from a given equation of a lineFrom the equation ax+by+c=0 >m=a/b >xintercept=c/a >yintercept=c/b
