Introduction to Solving Straight Line Equation

Introduction to Solving Straight Line Equation:

Straight line is a curve which has the same slope along its length. Basically  line is a series of points that extends in two opposite directions without end.There are many ways to solve the equations of straight lines. A straight line contains x-intercept, y-intercept and slope (m). An equation is a mathematical statement that two expressions are equal.
The general form of an equation of a straight line is
                                                     y = mx+b.
 Where m is the slope of the straight line and b is the intercept made by the straight line on y-axis.
If a Straight line passing trough two points (x₁,y₁) and (x₂,y₂) then slope of that line is `(y2-y1)/(x2-x1)`

• If two lines are parallel then their slopes are equal.
• If two lines are perpendicular to each other then product of their slopes is equal to -1.

There are different forms of equation of a straight line which are given below :

1. Slope-intercept form 
2. Intercept form 
3. Point-slope form 
4. Two-point form

The descriptions of each method will be explained below with an example. 

Different Forms of Equation of Straight Line:

1. Slope - Intercept form of a Straight Line:

Statement: The equation of the straight line with slope 'm' and Y-intercept 'c' is y = mx+c

Proof: Let L be the straight line whose slope is m and which cuts off an intercept 'c' on the Y-axis 

                   If P(x,y) is a point on the xy-plane, then

               P lies on L   => m = slope of L = (y-c)/(x-0)

                                      => y = mx+c

               Conversely, if P(x,y) satisfies the equation y = mx+c, then x = 0 => y = c.

2. Intercept form of a Straight Line:

Statement: The equation of the straight line which cuts off intercepts a and b on the X-axis and Y-axis respectively is`x/a+y/b=1`

Proof: The straight line L which cuts off intercepts a and b on the X-axis and Y -axis respectively meets these axis at A(a,0) and B(0,b) and therefore slope of the line = -b/a.

Hence, the equation of L , by the slope-intercept form is  `y= (-b/a)x +b` or `x/a+y/b=1` .

Example: Solve the Equation of straight line which makes intercepts whose sum is 5 and product is 6 

Solution : Let a and b are the intercepts made by the line , then 

given in the problem that a + b = 5 and ab = 6 

solving these equations we obtain a = 2 and b = 3 

Then, the required equation of the straight line is `x/2+ y/3=1` i.e 3x+2y-6 = 0

                                                                   or `x/3+ y/2=1` i.e 2x+3y-6 = 0

3. Point - Slope form of a Straight Line:

Statement: The equation of the straight line with slope m and passing through the point (x1,y1) is y - y1 = m(x-x1).

Proof: Equation of any straight line with slope m is of the form y = mx+c.

This line passes through the point (x1,y1) is y1 - mx1 = c

Therefore, the equation of the line with slope m containing the point (x1,y1) is 

         y - mx=c = y1 - m1x1

    i.e y-y1 = m(x-x1)

Example: Solve the equation of straight line which has the slope -1 and passes through the point (-2,3) 

Solution: The slope of the given straight line m = -1 

and the point on the line is ( -2,3) 

hence the equation of the line is 

                           y-3= -1(x+2) 

                           x+y-1=0.

4. Two-Point form of a Straight Line:

Statement: The equation of the straight line passing through the point A(x1,y1) and B(x2,y2) is

(x-x1)(y1-y2)=(y-y1(x1-x2).

Proof: Let L be the Straight line containing the points A(x1,y1) and B(x2,y2) 

Case 1: suppose L is  non-vertical. Then x1 ≠x2 and slope of L = `(y1-y2)/(x2-x1)`

Therefore, the equation of L is `y-y1 = (y1-y2)(x-x1)/ (x1-x2)`

 i.e  (x-x1)(y1-y2) = (y-y1)(x1-x2)     ……(1)

Case 2: If L is vertical , then x1=x2 and y1≠y1. Equation of L, in this case, is x = x1 [form (1)]

Example: Solve the equation of straight line which is passing through the point (1,-2) and (-2,3)

Solution: The slope of the line containing (1,-2) and (-2,3) is

               m= (3+2)/(-2-1) = -5/3

and hence equation is y+2=-5/3(x-1)

 i.e      5x+3y+1=0

Exercise:

Problem 1: The following equation are in slope-intercept form. In each case, specify the slope and  y-intercept.

i) y=2x+7   (Ans: slope = 2, y-intercept = 7)

ii)  y=−4x+2    (Ans: slope = -4, y-intercept = 2)    

iii)  2x+5y=15.    (Ans: slope = `-2/5` , y-intercept = 3)         

Problem 2: Find the equation of line passing through two points (2,3) and (5,6).

Ans: y = x+1.