## Algebra

Showing posts with label cartesian. Show all posts
Showing posts with label cartesian. Show all posts

### Support Course with Tracy Redden

Tracy Redden's support course.

1. A review of graphing lines, finding slope, and finding equations of lines from 2 points and a perpendicular line.

2. Review how to multiply and Factor all types of Polynomials.  I cover Common Factoring, Factor by grouping, Trinomial factoring, and difference and sum of cubes!

3. I show you how to solve all types of linear equations from  basic linear equations to more complicated ones with fractions and variables on both sides.  There are also the ones that end up with no solutions and all reals as a solution.

4. The first step into learning how to solve a quadratic is by factoring.  Here I show you how and explain why.

5. In addition we will look at the domains and restrictions of Rational Expressions.

6. How to Add and Subtract Rational Expressions.  I show you how to find common denominators so you can simplify.

7. How to simplify Complex Fractions.  I show you two different methods.

... more to come soon.

### The Rectangular Coordinate System

The rectangular coordinate system consists of two real number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. These two number lines define a flat surface called a plane, and each point on this plane is associated with an ordered pair of real numbers (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. The intersection of the two axes is known as the origin, which corresponds to the point (0, 0).

The x- and y-axes break the plane into four regions called quadrants, named using Roman numerals I, II, III, and IV. The ordered pair (x, y) represents the position of points relative to the origin. A few ordered pairs are plotted below:

This system is often called the Cartesian coordinate system, named after the French mathematician RenĂ© Descartes (1596–1650). Read about him on Wikipedia, he was a very interesting person!

A linear equation has standard form,
ax + by = c

where a, b, and c are real numbers and a and b are not both zero. With numbers a linear equation looks like,
6x  3y = 12
Solutions to linear equations are ordered pairs (x, y) where the coordinates, when substituted into the equation, produce a true statement. For example, to show that (3,2) and (−2, −8) are solutions substitute and then simplify:
Here we can see (3, 2) and (−2, −8) are solutions to the equation. From geometry we know that two points determine a line.  So at this point we can plot these points and then draw a line through them.  The line represents all ordered pair solutions and is called its graph.

We have graphed our first line!

Tip 1: The scale on the x-axis does not need to match the scale on the y-axis.  Choose a scale that is convenient. In this case, each tick on the y-axis represented 2 units. However, when you do this you need to be careful when plotting the points.

Tip 2: You do not really need graph paper, but it sure does help. Go to Google and search “free printable graph paper.” Or visit OpenAlgebra.com and click on the link to printable graph paper on the right and print some out.

Tip 3: Use a straightedge – a ruler or ATM card. All lines are straight, do not try to freehand them especially when accuracy is important.

Notice that (−3, 2) is not on the line and so these coordinates should not solve the equation.
Now, the question is, “how did we get those points?”  Read [ Graphing Lines by Plotting Points ] and you will see.
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### Rectangular Coordinate System

Sometimes referred to as the rectangular coordinate system, the Cartesian coordinate system consists of two perpendicular real number lines intersecting at zero. Positions on this grid system are identified using ordered pairs, (xy). The center of the system (0, 0) is called the origin.  The x-coordinate indicates horizontal distance from the origin and the y-coordinate indicates vertical distance from the origin.

The horizontal number line, usually called the x-axis, is typically used for the independent variable. The vertical real number line, usually called the y-axis, is used for the dependent variable.
Ordered pairs with 0 as a coordinate do not lie in a quadrant; these points lie on an axis.

Example: Find the distance and midpoint between the two points: (3, 4) and (−1, 2).

Example: Find the distance and midpoint between the two points: (0, 0) and (−3, 4).

Example: Find the distance and midpoint between the two points: (−1,−1) and (1,1).

Example: Find the distance and midpoint between the two points: (−2,−5) and (−4,−3).

Circle Word Problem: If the diameter of a circle is defined by two points (−3, 4) and (7, 4), find the center and radius of the circle.  (Hint: diameter = 2*radius)

Area of a Circle: Find the area of a circle given the center (−3, 3) and a point (3, 3) on the circle.