What you should learn

To write linear equations in point-slope form

To write linear equations in standard form

NCTM Curriculm Standards 2, 3, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Point-slope form

Standard form

Introduction: If you lived in Miami, Florida, and moved to Denver, Colorado, what adjustments do you think you might have to make? Obviously, the weather is different, but did you know you would also have to adjust to living at a higher altitude? As the altitude increases, the oxygen in the air decreases. This can affect your breathing and cause dizziness, headache, insomnia, and loss of appetite.

Over time, people who move to higher elevations experience acclimatization, or the process of getting accustomed to the new climate. Their bodies develop more red blood cells to carry oxygen to the muscles. Generally, long-term acclimatization to an altitude of 7000 feet takes about 2 weeks. After that, it takes 1 week for each additional 2000 feet of altitude. The graph below shows the acclimatization for altitudes greater than 7000 feet.

We can use any two points on the graph to find the slope of the line. For example, let (9000, 3) and (11,000, 4) represent (x1, y1) and (x2, y2), respectively.

m = (y2 - y1)/(x2 - x1) = (4 - 3)/(11,000 - 9000) = 1/2000

Suppose we let (x, y) represent any other point on the line. We can use the slope and one of the given ordered pairs to write an equation for the line.

m = (y2 - y1)/(x2 - x1)

1/2000 = (y - 3)/(x - 9000)

(1/2000)(x - 9000) = y - 3

y - 3 = (1/2000)(x - 9000)

Since this form of the equation was generated using the coordinates of a known point and the slope of the line, it is called the point-slope form.

Point-Slope Form of a Linear Equation: For a given point (x1, y1) on a nonvertical line having slope m, the point-slope form of a linear equation is as follows. y - y1 = m(x - x1)

 

You can write an equation in point-slope form for the graph of any nonvertical line if you know the slope of the line and the coordinates of one point on that line.

Exercise 1: Write the point-slope form of an equation for each line.

a. A line that passes through (-3, 5) and has a slope of -3/4

y - y1 = m (x - x1)

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

The equation of the line is y - 5 = (-3/4)(x + 3)

b. A horizontal line that passes through (-6, 2)

 

 

A vertical line has a slope that is undefined, so you cannot use the point-slope form of an equation. However, you can write the equation of a vertical line by using the coordinates of the points thorugh which it passes. Suppose a line passes through (3, 5) and (3, -2). The equation of the line is x = 3, since the x-coordinate of every point on the line is 3.

Any linear equation can also be expressed in the form Ax + By + C, where A, B, and C are integers and A and B are not both zero. This is called the standard form of a linear equation.

Standard Form of a Linear Equation: The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, A0, and A and B are not both zero.

 

Linear equations that are written in point-slope form can be rewritten in standard form.

Exercise 2: Write y + 5 = (-5/4)( x - 2) in standard form.

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

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

4y + 20 = -5(x - 2)

4y + 20 = -5x + 10

4y = -5x - 10

5x + 4y = -10

The standard form of the equation is 5x + 4y = -10

 

 

You can write an equation of a line if you know the coordinates of two points on that line.

Exercise 3: Write the point-slope form and the standard form of an equation of the line that passes through (-8, 3) and (4, 5)

Exercise 4: Write the equations of the lines containing the sides of parallelogram ABCD in point-slope and standard forms.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 19 - 47 odd, 48 - 55

Alternative Homework: Enriched: 18 - 44 even, 45 - 55

Extra Practice: Students book page 769 Lesson 6-2

Extra Practice Worksheet: Click Here.

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