FIND A MISSING COORDINATE USING SLOPE

Example 1 :

The slope of a line is 3/2 and the line contains the points (5, 9) and (3, a). What is the value of a ?

Solution :

Formula to find the slope of a line when two points are given : 

m  =  (y₂ - y₁) / (x₂ - x₁)

Given : Slope of the line is 3/2. 

Then, 

(y₂ - y₁) / (x₂ - x₁)  =  3/2

Substitute (x₁, y₁) = (5, 9) and (x₂, y₂) = (3, a). 

(a - 9) / (3 - 5)  =  3/2

(a - 9) / (-2)  =  3/2

Multiply each side by (-2).

a - 9  =  -3

Add 9 to each side. 

a  =  6

Example 2 :

The slope of a line is -2 and the line contains the points (7 , 4) and (x, 12). What is the value of x ? 

Solution :

Formula to find the slope of a line when two points are given : 

m  =  (y₂ - y₁) / (x₂ - x₁)

Given : Slope of the line is -2. 

Then, 

(y₂ - y₁) / (x₂ - x₁)  =  -2

Substitute (x₁, y₁) = (7, 4) and (x₂, y₂) = (x, 12). 

(12 - 4) / (x - 7)  =  -2

8 / (x - 7)  =  -2

Take reciprocal on each side. 

(x - 7) / 8  =  -1/2

Multiply each side by 8.

x - 7  =  -4

Add 7 to each side. 

x  =  3

Example 3 :

The slope of a line is 2/t and the line contains the points (-2 ,4) and (-6, 10). What is the value of t?

Solution :

Formula to find the slope of a line when two points are given : 

m  =  (y₂ - y₁) / (x₂ - x₁)

Given : Slope of the line is 2/t. 

Then, 

(y₂ - y₁) / (x₂ - x₁)  =  2/t

Substitute (x₁, y₁) = (-2, 4) and (x₂, y₂) = (-6, 10). 

(10 - 4) / (-6 + 2)  =  2/t

6 / (-4)  =  2/t

-3/2  =  2/t

Take reciprocal on each side. 

-2/3  =  t/2

Multiply each side by 2.

-4/3  =  t

Example 4 :

The line through the points (-2, a) and (9, 3) has slope -1/2. Find the value of a.

Solution :

Formula to find the slope of a line when two points are given : 

m  =  (y₂ - y₁) / (x₂ - x₁)

Given : Slope of the line is -1/2. 

Then, 

(y₂ - y₁) / (x₂ - x₁)  =  2/t

Substitute (x₁, y₁) = (-2, a) and (x₂, y₂) = (9, 3). 

(3 - a) / (9 + 2)  =  -1/2

(3 - a) / 11  =  -1/2

Multiply each side by 11.

3 - a  =  -11/2

Subtract 3 from each side.

-a  =  -11/2 - 3

-a  =  -11/2 - 6/2

-a  =  (-11 - 6) / 2

-a  =  -17/2

Multiply each side by (-1).

a  =  17/2

Example 5 :

The line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24) . Find the value of x.

Solution :

Slope of the line joining (-2, 6) and (4, 8) : 

m  =  (8 - 6)/(4 - (-2))

  =  2 / (4 + 2) 

  =  2/6

  =  1/3 -----(1)

Slope of the line joining (8, 12) and (x, 24) .

m  =  (24 - 12)/(x - 8)

  =  12/(x - 8) -----(2)

if lines are perpendicular to each other, the product of the slopes is equal to -1.

Then, 

(1/3) ⋅ 12/(x - 8) =  -1

4/(x - 8)  =  -1

4  =  -(x - 8)

4  =  -x + 8

x  =  8 - 4

x  =  4

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