01a cpt 1 lecture notes source-b - university of minnesota

L14-Wed-5-Oct-2016-Sec-1-3-Lines-HW13-Moodle-Q12 6

L14-Wed-5-Oct-2016-Sec-1-3-Lines-HW13-Moodle-Q12

"The Boxer" is a folk rock ballad written by Paul Simon in 1968

L14-Wed-5-Oct-2016-Sec-1-3-Lines-HW13-Moodle-Q12 Page 17


Today will discuss straight lines.

This is really an elementary algebra topic.

Suppose we have the following situation for a truck rental company:

Opening month special.

Rent our trucks for $10 down plus $2 per mile

We can construct a mathematical model for this situation that shows the relationship between the miles you drive and the cost of the rental:

x = miles driven

y = cost of rental in dollars.

First, write out in English how to solve the problem. Then, we translate this into an algebraic model.

Total Cost = Initial Cost + Cost for miles driven

The initial cost is given as $10. That is, you pay $10 even if you drive 0 miles.

The cost for miles driven is given determined by two things:

• by the rate $2 per mile = $2

1 mile

• and by the number of miles driven.

If we multiply the rate by the number of miles driven we get the cost for the miles driven.


We can make this a bit more realistic by including the sales tax, which is 7% of the total cost.

Now, the initial cost is 10.7 and the rate is 2.14 dollars per mile. You must also pay for gas. Let's say the truck gets 10 miles to the gallon and a gallon costs $3.

The additional cost for the gas is: 3 dollars 1 gallon 0.3 dollars per mile1 gallon 10 miles

•

So, the cost now is:

Why is there no tax on gas?


These are linear relations and can be generalized as y mx b . Knowing x we can predict y.

This is called the slope-intercept form of a line • x is the input or independent variable • y is the output or dependent variable • b is the y-intercept (sometimes called the initial value)

• m is the slope and is often called the rate of change; it tells us how the output changes in relation to changes in the input. For example, in the first equation y = 2x + 10, the rate of change is 2 dollars per mile. This rate of change tells us that the output changes by 2 dollars when the input changes by 1 mile.

Since the slope is the rate of change and tells us how the output changes in relation to the input we will define it this way:

2 1

2 1

riserun

y yymx x x

The delta means "change in". We define yx

as the average rate of change of y with respect to x.


There are two other forms of a line that we use, depending on the circumstances:


Use the definition of slope and find the slope between the point 1 1,x y and any other point, ,x y

1

1

1 1

ymx

y ymx x

m x x y y

This is the Point-slope form: 1 1y y m x x

where 1 1,x y is a particular point on the line

,x y is ANY other point on the line

m is the slope of the line.

So, for example, if you are told that the slope of a line is 2 and it passes through the point (3, -4) you can use this form to get an equation for the line.

1 1

4 2 3y y m x xy x

If we solve this for y we get the slope-intercept form of the line.

4 2 34 2 6

2 10

y xy x

y x

Here, we can see that the slope is 2 and if we put in the point (3, -4) we get a true statement.

where A, B, and C are real numbers and

A and B are both not 0.

We can put the above equation y = 2x - 10 into general form by moving the x term to the left side.

y = 2x - 10

-2x + y = -10

Notice that the b in y = mx + b and the B in Ax + By = C are not the same thing! A B is not always a B.

The General Form is often used when solving systems of equations.

Fixed point Variable point


1 5 4 143 1

1Use , to find

14

11 3

4

3

y mx b

m

y x bb

bb

y x


The x-value of each point is the same, 4. So, the equation is 4x

The y-value of each point is the same, -5. So, the equation is 5y


Parallel Lines:

Two non-vertical lines are parallel if and only if (IFF) their slopes are equal and they have different y-intercepts. This means their average rates of change are the same but their initial values are different.

Perpendicular Lines: Let's figure out the relationship between the slopes of two perpendicular lines:


2 3 63 2 6

2 2323

32

x yy x

y x

m

m

3235 22

2

y x b

b

b

3 22

y x

01a cpt 1 lecture notes source-b - university of minnesota

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