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MAT091 :: Lecture Note :: Week 10

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### Linear Functions

M-W.com defines linear as follows.

```   "of, relating to, resembling, or having a graph that
is a line and especially a straight line"

[...and...]

"having or being a response or output that is directly
proportional to the input"
```

Observe that the word linear contains the word line.

A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."

The following are equations for a line.

```   slope-intercept form:  y = mx + b
... used when slope and y-intercept are known

point-slope form:  y - y1 = m(x - x1)
... where (x1, y1) is a point on the line having slope m

standard form:  Ax + By = C
```

Slope is another way of saying "rate of change" and linear functions have a constant rate of change.

Slope is often described as the ratio "rise over run."

Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.

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### Slope (Linear Functions)

The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.

The following is how to find the slope between two unique points on a line.

```   point 1:  (x1, y1)
point 2:  (x2, y2)

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

note:  x1 ≠ x2
```

Slope represents the ratio of change-in-output over change-in-input.

```   m = Δy / Δx    (change in output / change in input)

Δ is the "delta" character
Δy = y2 - y1
```

Slope is often described as the ratio "rise over run."

```
rise
m = ------
run
```

Some slope notes.

```   horizonal lines have slope 0   (y = f(x) = k, where k is a consant)
vertical lines have undefined slope   (x = k, where k is a consant)
parallel lines have equal slopes     (m1 = m2)
perpendicular lines have negative inverse slopes  (m2 = -1/m1)
```

The following is a verbose form the slope-intercept equation.

```   output = slope * input + initial_value_if_any

...or...

output = average_rate_of_change * input + initial_value_if_any
```

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### Intercepts (Linear Functions)

We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonal-axis and vertical-axis (often called the x-intercept and y-intercept, respectively).

The vertical-intercept for a function is found by evaluating the function when its input is zero. The orderd-pair for a vertical-intercept will be `(0, something)`.

The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output. The orderd-pair for a horizonal-intercept will be `(something, 0)`.

```   f(x) = 5(x) + 3

vertical-intercept (y-intercept)...
set the input to 0 and evaluate:  f(0) = 5(0) + 3 = 3
vertical-intercept is (0, 3)

horizontal-intercept (x-intercept)...
set the output to 0 and solve:  0 = 5(x) + 3
-3 = 5(x)
-3/5 = x
horizontal-intercept is (-3/5, 0)
```

Generalizations.

```   given  f(x) = m(x) + b

vertical-intercept:  (0, b)        [notice input is 0]
horizontal-intercept:  (-b/m, 0)     [notice output is 0]
```

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### Exercise: Find Equation of a Parallel Line

Parallel lines never intersect (i.e. they don't have any points in common) and they have equal slopes.

Given the line `y = 10x + 2` find the equation of a parallel line that passes through the point `(5, 40)`.

```   y = 10x + b                    # parallel lines have the same slope
y - 40 = 10(x - 5)             # using point-slope form
y - 40 = 10x - 50              # apply distributive property
y - 40 + 40 = 10x - 50 + 40    # add 40 to both sides
y = 10x - 10                   # y = mx + b form

[check answer] substitute 5 in for x and we should get 40

y = 10(5) - 10 = 50 - 10 = 40
```

`[question]` No or Yes: The parallel line intercepts the vertical-axis at the point `(0, -10)`.

`[question]` No or Yes: The parallel line intercepts the horizontal-axis at the point `(1, 0)`.

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### Exercise: Find Equation of a Perpendicular Line

Perpendicular lines intersect (i.e. they have a point in common) and they form right angles at their point of intersection.

The slopes of perpendicular lines are negative reciprocals of each other.

```   m * mp = -1

'm' is the slope of a line and 'mp' is the slope of a perpendicular line
```

Given the line `y = 5x + 2`, find the equation of a perpendicular line that passes through the point `(5, 10)`.

```   Let A be the given line and B the perpendicular line.

slope of line A:  m = 5
slope of line B:  m = -(1/5)     # negative reciprocal

find the slope-intercept equation for line B

substitute point (5, 10) into point-slope equation...
y - 10 = -(1/5)(x - 5)

solve for y...
y - 10 + 10 = -(1/5)(x - 5) + 10     # add 10 to both sides
y = -(1/5)(x - 5) + 10               # simplify
y = -(1/5)x - -(1/5)5 + 10           # distribute -(1/5)
y = -(1/5)x + 1 + 10                 # simplify
y = -(1/5)x + 11
```

`[question]` No or Yes: The perpendicular line intercepts the vertical-axis at the point `(0, 11)`.

`[question]` No or Yes: The perpendicular line intercepts the horizontal-axis at the point `(55, 0)`.

`[exercise]` Find the point where lines A and B intersect.

```   set line A equal to line B
5x + 2 = -(1/5)x + 11

solve for x...
5x + 2 = -0.2x + 11             # divide 1 by 5
5x + 2 - 2 = -0.2x + 11 - 2     # subtract 2 from both sides
5x = -0.2x + 9                  # simplify
5x + 0.2x = 9                   # add 0.2x to both sides
5.2x = 9                        # simplify
5.2x / 5.2 = 9 / 5.2            # divide both sides by 5.2
x = 1.73                        # simplify

substitute x in one of the equations...
y = 5(1.73) + 2 = 10.65

point of intersection:  (1.73, 10.65)

check using the other line...
-(1/5)1.73 + 11 = 10.65
```

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### Intersecting Lines

Two lines that are not parallel can intersect. If the equations for the lines are known, then the point of intersection can be determined as follows.

```   line A:  y = 2x + 2
line B:  y = 5x + 5

set the lines equal to one another...
2x + 2 = 5x + 5

solve for x...
2x + 2 + -2 = 5x + 5 + -2    # add -2 to both sides
2x = 5x + 3

2x + -5x = 5x + 3 + -5x      # add -5x to both sides
-3x = 3

-3x / -3 = 3 / -3            # divide both sides by -3
x = -1

substitute for x in one of the line equations and evaluate...
y = 2(-1) + 2 = -2 + 2 = 0

point of intersection:  (-1, 0)

check answer: plug (-1, 0) into the other line equation
and see if it works...
y = 5(-1) + 5 = -5 + 5 = 0
```

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### Example Application Using Intersecting Lines

Zelmo Zeroman had to select between two linear pay scales. One choice was to receive \$10 up-front and then earn \$1 for every 8 hours worked. The second choice was to received \$2 up-front and then earn \$1 for every 5 hours worked. The following two functions define Zelmo's pay scale choices.

```           1                                  1
p(h) = ---(h) + 10                 P(h) = ---(h) + 2
8                                  5

h:  number of hours
p(h):  USD
P(h):  USD

The amount of pay is a function of hours worked.
The amount of pay depends on the number of hours worked.
```

Zelmo realized that over time the `P(h)` pay scale would result in more pay, but he wanted to learn how many hours he needed to work before `P(h)` would be greater than `p(h)`.

It was suggested that Zelmo find where `P(h)` and `p(h)` intersect when graphed.

```   observe...  1/8 = 0.125    and    1/5 = 0.2

p(h) = 0.125(h) + 10       and    P(h) = 0.2(h) + 2

let p(h) = P(h)...

0.125(h) + 10 = 0.2(h) + 2

solve for 'h'...

...subtract 0.125(h) from both sides...
-0.125(h) + 0.125(h) + 10 = -0.125(h) + 0.2(h) + 2
10 = 0.075(h) + 2

...subtract 2 from both sides...
-2 + 10 = 0.075(h) + 2 - 2
8 = 0.075(h)

...divide both sides by 0.075...
8/0.075 = 0.075(h)/0.075
106.67 = h
```

Conclusion: Zelmo should pick the `p(h)` pay scale if he is going to work less than `106.67 hours`; otherwise, he should select the pay scale defined by `P(h)`.

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### Zelmo Zeroman Borrows Money From Herb Mumford

Zelmo Zeroman borrowed \$1,000 from his "friend" Herb Mumford. Zelmo and Herb agreed that the loan was to be paid off at a rate of \$200 per week. Zelmo created the following function to help him keep track of his weekly payoff amounts.

```   p(w) = 1000 - 200(w) USD

input 'w' is week number  and  USD stands for U.S. Dollar

domain:  w ≥ 0, where 'w' is an integer
range:  0 ≤ p(w) ≤ 1000
```

The function p(w) outputs the payoff amount after w weeks.

##### Exercises
1. Re-write the function in y = mx + b form.

2. p(2) = _________   [record unit with answer]

3. The function p(w) has a slope of `_______`.

4. Explain why p(w) is a decreasing function.

5. It will take Zelmo ________ weeks to payoff the loan.

6. In this particular case, the technique used to answer the previous exercise is also the technique used to find the _______-intercept for a function.

7. Graph p(w) for the entire duration of the loan.

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### Inequalities

The following are some inequalities graphed on the number line. An end-point marked with 'x' (or ']') implies that value is included, while an end-point marked with 'o' (or ')') is excluded.

```   x > 2

---------------------o=========>
-5 -4 -3 -2 -1  0  1  2  3  4  5

x ≥ -4

---x===========================>
-5 -4 -3 -2 -1  0  1  2  3  4  5

x ≤ 3

<=========================x------
-5 -4 -3 -2 -1  0  1  2  3  4  5

x < -2

<==========o---------------------
-5 -4 -3 -2 -1  0  1  2  3  4  5
```

Linear inequalities are solved just like linear equations with one exception: the inequality sign is "flipped" whenever there is a multiply (or divide) by a negative. The term "flipped" is used to imply less-than becomes greater-than and vice versa; and that less-than-or-equal-to becomes greater-than-or-equal-to and vice versa.

```   Example 1:  8x + 1 < 2x - 5

8x - 2x + 1 < -5
6x + 1 - 1 < -5 - 1
6x < -6
x < -1

try x = 0...
8(0) + 1 = 9; 2(0) - 5 = -5
9 is not less than -5

try x = -1...
8(-1) + 1 = -7; 2(-1) - 5 = -7
-7 is not less than -7

try x = -2...
8(-2) + 1 = -15; 2(-2) - 5 = -9
-15 is less than -9

Example 2:  -2x < 5

-2x / -2 > 5 / -2      # < flipped to > (divide-by negative)
x > -2.5

try x = 5...
-2(5) < 5
-10 < 5 is true

try x = -5...
-2(-5) < 5
10 < 10 is false
```

The following uses numbers to help understand when inequalities are flipped (switched).

```   10 > 5  is true
multiply both sides by -1
10(-1) = -10  and  5(-1) = -5
-10 > -5 is false
```

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### Systems of Linear Equations

A "system" of equations is a collection of equations that are treated as an atomic unit. In other words, a system of equations is used when dealing with more than one variable.

```   A: 3x + 3y = 0
B: 12x + 2y = -20

pick one of the equations...
pick A

solve equation A for y
3x + 3y = 0
-3x + 3x + 3y = 0 + -3x
3y = -3x
3y / 3 = -3x / 3
y = -x

substitute for y in the equation B...
12x + 2(-x) = -20

solve equation B for x...
12x + -2x = -20
10x = -20
10x / 10 = -20 / 10
x = -2

pick one of the equations...
pick A

substitute for x in the equation A...
3(-2) + 3y = 0

solve equation A for y...
3(-2) + 3y = 0
-6 + 3y = 0
6 + -6 + 3y = 0 + 6
3y = 6
3y / 3 =  6 / 3
y = 2

the solution is the point (-2, 2)
```

Another system of linear equations.

```   Two numbers have a sum of 55 and a difference of 9.
Find the numbers.

let x and y be the two unknown numbers

form two equations...
A: x + y = 55
B: x - y = 9

x + y = 55
+  x - y =  9
=============
2x = 64

solve for x...
2x = 64
2x / 2 = 64 / 2
x = 32

pick an equation, substitute for x, solve for y
pick A
32 + y = 55
-32 + 32 + y = 55 + -32
y = 23

solution:  (32, 23)

[exercise] Re-do this problem using substitution.
```

Here is another system of linear equations that is solved using the addition method.

```   A: 5x + y = 10
B:  x + y = 20

multiply both sides of B by -1...
-1(x + y) = 20(-1)
-x - y = -20

5x + y =  10
+ -1x - y = -20
===============
4x     = -10

solve for x...
4x / 4 = -10 / 4
x     = -2.5

pick an equation and solve for y...
pick B
-2.5 + y = 20
y = 22.5

solution:  (-2.5, 22.5)

subtitute the solution into A and see if it works...
5(-2.5) + 22.5 = 10
-12.5 + 22.5 = 10
it works!
```

Is a system of equations necessary to solve the following problem?

```   Ten less than half of a number is thirty. What is the number?
```

No, because there is only one variable.

```   n/2 - 10 = 30
n/2 - 10 + 10 = 30 + 10
n/2 = 40
n/2 * 2 = 40 * 2
n = 80
```

Here is a exercise that can be solved using a system of equations.

```   A new nightclub had a grand opening in Scottsdale.  First night
admission was \$4 for females and \$6 for males.  90 people attended
the grand opening and the nightclub collected \$460.  How many
females and how many males attended the grand opening?
```
```   let 'f' represent # of females and let 'm' represent # of males

A: f + m = 90
B: \$4(f) + \$6(m) = \$460

pick A and solve for 'f'...
f = 90 - m

substitute 'f' into B...
4(90 - m) + 6m = 460

distribute the 4, combine like-terms, solve for 'm'...
360 - 4m + 6m = 460
360 + 2m = 460
-360 + 360 + 2m = 460 - 360
2m = 100
(1/2)2m = 100(1/2)
m = 50

substitute m = 50 into A and solve for 'f'...
f + 50 = 90
f = 40

There were 40 females and 50 males the grand opening.
```

Education.Yahoo.com:: Solving Linear Systems

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### Solving System of Linear Equations Using TI-83

The "intersection" feature of the TI-83 calculator can be used to find the solution to a system of linear equations.

```   press Y=
Y1 press CLEAR, if necessary
enter equation:  X is entered using key labeled X,T,theta,n
-2X + 3 ENTER
cursor at Y2, press CLEAR if necessary
enter equation:
(1/3)X - 4 ENTER
press GRAPH
press 2ND TRACE 5 (intersect)
First curve? ENTER
Second curve? ENTER
Guess? ENTER
Intersection x = 3 and y = -3   (3, -3)
```

Substitute `x = 3` into both equations and test if `(3, -3)` is a solution.

```   f(x) = -2x + 3
f(3) = -2(3) + 3 = -6 + 3 = -3     check

f(x) = (1/3)x - 4
f(3) = (1/3)(3) - 4 = 1 - 4 = -3   check
```

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