WRITE THE EQUATION OF A LINE IN STANDARD FORM WITH ONE POINT AND SLOPE

Example 1 :

Find the equation of the straight line whose.

(i)  Slope is -4 and passing through (1, 2).

(ii)  Slope is 2/3 and passing through (5, -4).

Solution :

Part (i) :

To find the equation of the straight line, we have to use the formula of slope point form.

Slope (m) = -4, x₁ = 1 and y₁ = 2.

Equation of the line  :

(y - y₁) = m(x - x₁)

(y – 2) = -4(x – 1)

y – 2 = -4x + 4

4x + y – 2 - 4 = 0

4x + y - 6 = 0

Part (ii) :

To find the equation of the straight line we have to use the formula of slope point form.

Slope (m) = 2/3, x₁ = 5 and y₁ = -4.

Equation of the line  :

(y - y₁) = m(x - x₁)

y – (-4) = (2/3)(x – 5)

 (y + 4) = (2/3)(x – 5)

3(y + 4) = 2(x – 5)

3y + 12 = 2x – 10

2x – 3y - 10 - 12 = 0

2x – 3y – 22 = 0

Example 2 :

Find the equation of the straight line which passes through the midpoint of the line segment joining

(4, 2) and (3, 1)

whose angle of inclination is 30 degree.

Solution :

First we have to find midpoint of the line segment joining the points (4, 2) and (3, 1).

Midpoint = (x₁ + x₂)/2, (y₁ + y₂)/2

= (4 + 3)/2, (2 + 1)/2

= (7/2, 3/2)

angle of inclination = 30°.

θ = 30°

Slope (m) = tan θ

m = tan 30°

m = 1/√3

Equation of the line :

y - y₁ = m(x - x₁)

y - 3/2 = (1/√3)(x - 7/2)

2y - 3 = (1/√3)(2x - 7)

2√3y - 3√3 = 2x - 7

2x - 2√3y - 7 + 3√3 = 0

Example 3 :

Excluding hydropower, U.S. power plants used renewable energy sources to generate 105 million megawatt hours of electricity in 2007. By 2012, the amount of electricity generated had increased to 219 million megawatt hours.

Write a linear model that represents the number of megawatt hours generated by non-hydropower renewable energy sources as a function of the number of years since 2007. Use the model to predict the number of megawatt hours that will be generated in 2017.

Solution :

Let x represent the time (in years) since 2007 and let y represent the number of megawatt hours (in millions). Because time x is defined in years since 2007,

2007 corresponds to x = 0 and 2012 corresponds to x = 5.

Let (x₁, y₁) = (0, 105) and (x₂, y₂) = (5, 219). The initial value is the y-intercept b, which is 105. The rate of change is the slope m.

Slope = (219 - 105) / (5 - 0)

= 114/5

= 22.8

y = mx + b

y = 22.8x + 105

To find the number of megawatt hours that will be generated in 2017, we have to find the difference between 2007 and 2017, then it is 10.

Applying 10, we get

y = 22.8(10) + 105

y = 228 + 105

= 333 megawatt.

Example 4 :

In 1960, the world record for the men’s mile was 3.91 minutes. In 1980, the record time was 3.81 minutes. 

a. Write a linear model that represents the world record (in minutes) for the men’s mile as a function of the number of years since 1960.

b. Use the model to estimate the record time in 2000 and predict the record time in 2020.

Solution :

a)

Initial value = 3.91

Let x be the number of years since 1960.

y be the world record for the men’s mile

x₁ = 0 (in the year 1960), y₁ = 3.91

x₂ = 20 (in the year 1980), y₁ = 3.81

Slope = (3.81 - 3.91) / (20 - 0)

= -0.1/20

= -0.005

y = -0.005x + b

y = -0.005x + 3.91

b)

In 2000, then x = 40

y = -0.005(40) + 3.91

y = -0.2 + 3.91

y = 3.71

In 2010, then x = 50

y = -0.005(50) + 3.91

y = -0.25 + 3.91

y = 3.66

Example 5 :

A website hosting company charges an initial fee of $48 to set up a website. The company charges $44 per month to maintain the website.

a. Write a linear model that represents the total cost of setting up and maintaining a website as a function of the number of months it is maintained.

b. Find the total cost of setting up a website and maintaining it for 6 months.

Solution :

a) Initial fee = $48

Charge per month = $44

Let x be the number of months and y be the total cost.

y = 44x + 48

b) Total cost for 6 months

y = 44(6) + 48

= 264 + 48

312

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