EQUATIONS OF STRAIGHT LINES

In this section, you will learn about the equations of straight lines.

These equations can take various forms depending on the facts we know about the lines. So to start, suppose we have a straight line containing the points in the following list.

There are many more points on the line, but we have enough now to see a pattern. If you take any x value and add 2, we get the corresponding y value. 

That is, 

0 + 2  =  2

1 + 2  =  3

2 + 2  =  4

3 + 2  =  5

Here, we can see a fixed relationship between the x and y co-ordinates of any point on the line, and the equation

y  =  x + 2

is always true for points on the line. We can label the line using this equation.

Suppose that we have the graph of a straight line and we want to find its equation.

For any straight line, if we want to find the equation, we must have the following information of that straight line. 

(i)  Slope and y-intercept

(ii)  One point and slope

(iii) Two points

(iv) Two intercepts (x-intercept and y-intercept)

If we have any one of the five information given above we will be able to find the equation of a straight line using the formulas given below. 

Now, let us look at the different forms equation of a straight line.

Different Forms of Equations of Straight Lines

1. Slope-Intercept form equation of a line :

y = mx + b

m ---> slope

b ----> y-intercept

2. Point-Slope form equation of a line :

y - y₁ = m(x - x₁)

m ---> slope

(x₁, y₁) ----> point

3. Two-Points form equation of a line :

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

Two points ----> (x₁, y₁) and (x₂, y₂)

4. Intercept-Form equation of a line :

x/a + y/b = 1

a ----> x-intercept

b ----> y-intercept

Apart from the above forms of equation of straight line, there are some other ways to get equation of a straight line. 

1. If a straight line is passing through a point (0, k) on y-axis and parallel to x-axis, then  the equation of the straight line is  

y  =  k

2. If a straight line is passing through a point (c, 0) on x-axis and parallel to y-axis, then  the equation of the straight line is  

x  =  c

3. Equation of x-axis is 

y  =  0

(Because, the value of 'y' in all the points  on x-axis is zero)

4. Equation of y-axis is  

x  =  0

(Because, the value of 'x' in all the points  on y-axis is zero)

5. General form of equation of a straight line is

ax + by + c  =  0

Solved Problems

Problem 1 :

Find the general form of equation of a straight line whose slope is 3 and y-intercept -2. 

Solution :

Given : Slope  m = 3 and y-intercept b = -2.

Equation of the straight line in slope-intercept form : 

y  =  mx + b

Substitute m = 3 for m and b = -2.

y  =  3x - 2

Subtract y from each side.

0  =  3x - y - 2

or

3x - y - 2  =  0

Problem 2 :

Find the general form of equation of a straight line passing through the points (-1, 1) and (2, -4). 

Solution :

Given : Two points on the straight line : (-1, 1) and  (2, -4).

Equation of the straight line in two-points form is 

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

Substitute (x₁ , y₁)  =  (-1, 1) and (x₂, y₂)  =  (2, -4).

(y - 1) / (-4 - 1)  =  (x + 1) / (2 + 1)

Simplify.

(y - 1) / (-5)  =  (x + 1) / 3

Cross multiply.

3(y - 1)  =  -5(x + 1)

3y - 3  =  -5x - 5

5x + 3y + 2  =  0

Problem 3 :

Find the general equation of the straight line passing through the point (-2, 3) with slope 1/3.

Solution :

Given : Point  =  (-2, 3)  and  slope  m  =  1/3

Equation of the straight line in point-slope form is

y - y₁  =  m(x - x₁)  

Substitute (x₁ , y₁) = (-2 , 3) and m = 1/3. 

y - 3  =  1/3 ⋅ (x + 2)

Multiply each side by 3.

3(y - 3)  =  x + 2

Simplify. 

3y - 9  =  x + 2

Subtract 3y from each side. 

-9  =  x - 3y + 2

Add 9 to each side.

0  =  x - 3y + 11

or

x - 3y + 11 = 0

Problem 4 :

Find the general equation of the straight line whose x-intercept -2 and y-intercept is 3.

Solution :

Given : x-intercept is -2  and y-intercept is 3.

Equation of the straight line in intercept-form is 

x/a + y/b  =  1

Substitute a = -2 and b = 3. 

x/(-2) + y/3  =  1 -----(1)

The lest common multiple of (2, 3) is 6.

So, multiply each side (1) by 6. 

-3x + 2y  =  6

Multiply each side by -1.

3x - 2y  =  -6

Add 6 to each side. 

3x - 2y + 6  =  0

Problem 5 :

Find the equation of a straight line parallel to y-axis and passing through (-5, 0).

Solution :

Equation of a straight parallel to y-axis is 

x  =  c

It is passing through the point (-5, 0)

Then, 

-5  =  c

So, the equation of the given line is 

x  =  -5

or

x + 5  =  0

Problem 6 :

Find the equation of a straight line parallel to x-axis and passing through (0, 6).

Solution :

Equation of a straight parallel to x-axis is 

y  =  k

It is passing through the point (0, 6)

Then, 

6  =  k

So, the equation of the given line is 

y  =  6

or

x + 5  =  0

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