FIND THE EQUATION OF A LINE GIVEN A POINT AND THE SLOPE

How to find the equation of a line given a point and the slope ?

If given a point  (x₁, y₁) and a slope m, then we can use the below formula to find the equation of the line.

Formula for point-slope form :

y – y₁  =  m(x – x₁)

where,

(x₁, y₁) are the point on the line.

m  =  Slope of the line.

Find the equation of the line through :

Problem 1 :

(1,-2) having a gradient of 3

Solution :

Given, point (1, -2) and slope 3

Using point slope form formula :

Here x₁  =  1, y₁  =  -2 and m  =  3

y – y₁  =  m(x – x₁)

(y – (-2))  =  3(x – 1)

y + 2  =  3x – 3

y  =  3x – 3 – 2

y = 3x - 5

So, the required equation of a line is y = 3x - 5

Problem 2 :

(-4, -1) having a gradient of -2

Solution :

Given, point (-4, -1) and slope -2

Using point slope form formula :

Here x₁  =  -4, y₁  =  -1 and m  =  -2

y – y₁  =  m(x – x₁)

(y – (-1))  =  -2(x – (-4))

y + 1 =  -2(x + 4)

y + 1  =  -2x – 8

y  =  -2x – 8 – 1

y  =  -2x – 9

So, the required equation of a line is y  =  -2x – 9

Problem 3 :

(5, -2) having a gradient of -3

Solution :

Given, point (5, -2) and slope -3

Using point slope form formula :

Here x₁  =  5, y₁  =  -2 and m  =  -3

y – y₁  =  m(x – x₁)

(y – (-2))  = - 3(x – 5)

y + 2  =  -3x + 15

y  =  -3x + 15 – 2

y  =  -3x + 13

So, the required equation of a line is y  =  -3x + 13

Problem 4 :

(5, 2) having a gradient of 1/3

Solution :

Given, point (5, 2) and slope 1/3

Here x₁  =  5, y₁  =  2 and m  =  1/3

y – y₁  =  m(x – x₁)

y – 2 = 1/3(x – 5)

3y – 6  =  x - 5

3y  =  x – 5 + 6

3y  =  x + 1

y  =  x/3 + 1/3 

Problem 5 :

(-2, 8) having a gradient of -1/4

Solution :

Given, point (-2, 8) and slope -1/4

Here x₁  =  -2, y₁  =  8 and m  =  -1/4

y – y₁  =  m(x – x₁)

y – 8  =  -1/4(x – (-2))

y – 8  =  -1/4(x + 2)

4y – 32  =  -x – 2

4y  =  -x – 2 + 32

4y  =  -x + 30

y =  -x/4 + 30/4

y  = -x/4 + 15/2

Problem 6 :

(7, -3) having a gradient of 0

Solution :

Given, point (7, -3) and slope 0

y – y₁  =  m(x – x₁)

(y – (-3)) = 0(x – 7)

y + 3  =  0

y  =  -3

Problem 7 :

(1, 6) with gradient 2/3

Solution :

Given, point (1, 6) and slope 2/3

y – y₁  =  m(x – x₁)

y – 6 = 2/3(x – 1)

3y - 18 =  2(x – 1)

3y – 18  =  2x – 2

3y  =  2x – 2 + 18

3y =  2x + 16

y  =  2x/3 + 16/3   

Problem 8 :

(-5, 4) with gradient 3/5

Solution :

Given, point (-5, 4) and slope 3/5

y – y₁  =  m(x – x₁)

y – 4  = 3/5(x – (-5))

5y - 20 =  3(x + 5)

5y – 20  =  3x + 15

5y  =  3x + 15 + 20

5y =  3x + 35

y  = 3x/5 + 35/5

y  =  3x/5 + 7

Problem 9 :

(8, 0) with gradient -1/4

Solution :

Given, point (8, 0) and slope -1/4

y – y₁  =  m(x – x₁)

y – 0  =  -1/4(x – 8)

y =  -1/4(x - 8)

4y =  -x + 8

y =  -x/4 + 8/4

y  = -x/4 + 2

Problem 10 :

(8, -2) with gradient -3/4

Solution :

Given, point (8, -2) and slope -3/4

y – y₁  =  m(x – x₁)

(y – (-2))  = - 3/4(x – 8)

y + 2  =  -3/4(x -8)

4y + 8  =  -3x + 24

4y  =  -3x + 24 - 8

4y  =  -3x + 16

y  =  -3x/4 + 16/4

y  =  -3x/4 + 4

Problem 11 :

(-2, -4) with gradient 3

Solution :

Given, point (-2, -4) and slope 3

y – y₁  =  m(x – x₁)

(y – (-4))  =  3(x – (-2))

y + 4 =  3(x + 2)

y + 4  =  3x + 6

y  =  3x + 6 – 4

y  =  3x + 2

Problem 12 :

(5, -1) with gradient -5

Solution :

Given, point (5, -1) and slope -5

y – y₁  =  m(x – x₁)

(y – (-1))  = - 5(x – 5)

y + 1  =  -5x + 25

y  =  -5x + 25 - 1

y  = -5x + 24 

Problem 13 :

The graph of the a line in the xy-plane has slope 2 and contains the point (1, 8). The graph of a second line passes through the points (1, 2) and (2, 1). If the two lines intersect at the point (a, b) what is the value of a + b ?

a)  4    b)  3    c)  -1   d) -4

Solution :

Slope of the line = 2

Point on the line is (1, 8)

Equation of line :

(y - 8) = 2(x - 1)

y - 8 = 2x - 2

y = 2x - 2 + 8

y = 2x + 6 -----(1)

Equation of the line joining the points (1, 2) and (2, 1).

Slope = (1 - 2) / (2 - 1)

= -1/1

= -1

Equation of the line :

y - 2 = -1(x - 1)

y = -1x + 1 + 2

y = -x + 3 -----(2)

(1) = (2)

2x + 6 = -x + 3

2x + x = 3 - 6

3x = -3

x = -1

 Applying x = -1, y = -(-1) + 3

y = 4

Point of intersection is at (-1, 4)

a = -1 and b = 4

a + b = -1  + 4

a + b = 3

So, option b is correct

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.