WRITE THE EQUATION OF A LINE WITH GIVEN CHARACTERISTICS

Problem 1 :    

Find the equation of the line in general form, through:

(1, -5) with gradient 2/3

Solution :

Since we have a point and slope of the line, we use the formula 

y - y₁ = m(x - x₁)

to find the equation of the line.

Substitute (x₁, y₁) ==> (1, -5) and m = 2/3.

y + 5 = 2/3 (x - 1)

Multiply each side by 3

3(y + 5) = 2(x - 1)

3y + 15 = 2x - 2

2x - 3y = 15 + 2

2x - 3y = 17

Problem 2 :    

Find the equation of the line in general form through:

  (2, -3) and (-4, -5)

Solution :

Since we have two point on the line, we use the formula 

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

Substitute (x₁, y₁) = (2, -3) and (x₂, y₂) = (-4, -5)

(y + 3) / (-5 + 3) = (x - 2) / (-4 - 2)

(y + 3) / (-2) = (x - 2) / (-6)

Cross multiply,

(-6) (y + 3) = (-2) (x - 2)

-6y - 18 = -2x + 4

-6y + 2x = 18 + 4

2x - 6y = 22

Divide both sides by 2

x - 3y = 11

Problem 3 :

If 5x - 7y = 8 and 3x + ky = -11 are the equations of two lines, find the value of k for which:

a)   The lines are parallel

b) the lines are perpendicular         

Solution :

a)    5x - 7y = 8 and 3x + ky = -11

Slope of parallel lines are equal,

Slope of first line  =  5/7 ---(1)

Slope of the second line = -3/k ---(2)

(1) = (2)

-3/k = 5/7

k = -21/5

b)   Slope of general equation of line

ax + by + c = 0  is m = - coefficient of x / coefficient of y

Therefore,

Slope of line 5x - 7y = 8 is 5/7

Slope of line 3x + ky = -11 is -3/k

Now we know that two lines are perpendicular if their product of slopes is -1

5/7 × -3/k = -1

k = 15/7

Problem 4 :

Find the equation of the line in general form for the line shown below.

Solution :

Slope of the line = Rise / Run

Slope (m) = -1/3

Since it is falling line, will have negative slope.

m = -1/3

y - Intercept  :

The y - intercept of the line is the value of y where the line intersects the y- axis.

The above line intersects the y-axis at 1.

So,

y - Intercept (b) = 1

Equation of the line :

y = mx + b

y = (-1/3)x + 1 

Problem 5 :

Find the equation of the line in general form for the line shown below.

Solution :

Slope of the line = Rise / Run

Slope (m) = 1/2

Since it is raising line, will have positive slope.

m = 1/2

y - Intercept :

y - Intercept (b) = -1

Equation :

y = mx + b

y = (1/2)x - 1

Problem 6 :

Find the equation of the line in general form for the line shown below.

Solution :

Slope of the line = Rise / Run

Slope (m) = 2/2

Since it is falling line, will have negative slope.

m = -1

y - Intercept (b) = 2

Equation of the line :

y = mx + b

y = -x + 2

Problem 7 :

Find the equation of the line in general form for the line shown below.

Solution :

Slope :

m = (y₂ - y₁) / (x₂ - x₁)

Substitute (x₁, y₁) = (-1, 3) and (x₂, y₂) = (0, 0)

m = (0 - 3) / (0 + 1)

m = -3/1

m = -3

y - Intercept :

y - Intercept (b) = 0

Equation of the line :

y = mx + b

y = -3x + 0

y = -3x