# Miscellaneous Math Topics

**4 Equation of a Line**

A line is identified either by knowing any two points on the line or by knowing

one point and the slope (steepness) of the line.

We usually use Cartesian coordinates to plot lines . y is the vertical axis and

extends infinitely up ( positive ) and down ( negative ). x is the horizontal axis
and

extends infinitely left (negative) and right (positive). A point is represented
by

a pair of numbers (x,y) that provides its horizontal and vertical location.
There

is only one possible line that passes through any two points.

The slope of a line is its ratio of vertical to horizontal change (“rise” over

“run”). More formally, if and
are points on a line, the slope

is:

The order of the points does not matter as long as you are
consistent and use

the same order for both the numerator and denominator . For any two points on

a line you will get the same slope. The slope is often represented by the letter

m.

Excluding horizontal and vertical lines, every line
crosses both axes some-

where. The point where a line crosses the y-axis is called the y- intercept and
is

often represented by the letter b. The point where it crosses the x-axis is
called

the x-intercept. Horizontal lines cross the y-axis but not the x-axis and
vertical

lines cross only the x-axis.

Every line can be expressed by a simple equation in terms of x and y. Point

slope form is a common form for the equation of a line:

x and y are variables representing the points on the line,
is any point

on the line, and m is the slope. For example:

This is the line through the point (−3, 2) with the slope
. Another popular

form is the slope-intercept form:

y = mx + b

In this form, x and y are variables and represent some point on the line, m is

the slope and b is the y-intercept. For example, the equation of a line passing

throught the point (1, 2) with slope 3 is:

y = 3x − 1

I found the y-intercept by plugging the coordinates of the point (1, 2) into the

formula : y = 3x + b to get 2 = 3 + b and solving for b. To find another point

on the line, chose a value for x or y and solve for the other coordinate. The y

value when x = 2 is y = 3 * 2−1 = 5 so the line passes through the point (2, 5).

**4.1 Exercises: Equation of a line
**

1. A line passes through the points (1,3) and (2,0). What is the slope? Write

an equation for the line in slope-intercept form. What is the y-intercept?

What is the x-intercept?

2. A line has slope 2 and x-intercept 3. Write an equation for the line.

**5 Solutions to Exercises **

**1.2 Notation**

1. [−∞, 2]

2. 3

**1.3 Lists**

1. 30

1.4 Addition and Multiplication

1. 30

2. 120

3. 0

**1.6 The Factorial **

1. There are 3! ways. 3! = 3 * 2 * 1 = 6.

2. There are 4(3!) ways. 4 * 3! = 24

3. There are 3! ways to order the methods books and 4! ways to order the

theory books. There are 2 ways to put the books on the shelf (theory

on the left and methods on the right or vice versa). This is (3!)(4!)(2) =

(6)(24)(2) = 288.

**2.1 Exponents **

**2.2 Logarithms **

1.

Check this: 2^{2} = 4

2.

Check this: 3^{4} = 81

**3.1 Polynomials**

1. The order of 3x^{4} + 18x − 21 is 4.

2. (3x^{4} + 18x − 21) + (2x^{3} + 3y^{2} − 9x + 6) = 3x^{4}
+ 2x^{3} + 3y^{2} + 9x − 15

**3.2 Solving Polynomials
**

Find x in the equations below:

1. 2x − 4 = 0; x = 2

2. x

^{2}− 5x − 6 = 0 = (x − 6)(x + 1), x = 6 or x = −1

3. 3x^{2} − 10x + 7

Plug this into the quadratic formula :

So, or

Check this: 3(1^{2}) − 10(1) + 7 = 0. Check
yourself.

**4.1 Equation of a line
**

1. First, find the slope: or . Now we have most

of the equation: y = -3x + b. Second, find the y-intercept by plugging in

one of the points: (3) = (−3)(1)+b; b = 6. Check this by plugging in the

other point: 0 = (−3)(2) + 6 = 0. The equation is y = -3x + 6.

2. We know the slope, so we have y = 2x+b. The x-intercept is 2 so the line

passes through the point (2,0). Plugging in this point gives: 0 = 2(2) + b

so b = −4 and the equation is y = 2x − 4.

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