Algebra, Done!

I just finished all of algebra!

The last thing I did in algebra was conic sections.

In conics we learned how to graph ellipses, hyperbolas, circles, and parabolas.

This a line:

y = x + 5

Parabola:

y = -3x² - 6x - 1

Circle:

(x - 2)² + (y + 3)² = 9

Ellipse:

(x²/6) + (y²/16) = 1

Hyperbola:

x² - y² = 39

A parabola is U-shaped. But it can open up or down.

An ellipse is an oval.

Hyperbola is like two parabolas.

You can tell what shape an equation is just by looking at it.

If one variable either ,x or y, is squared it is a parabola.

If both variables are squared then the graph might be a circle, an ellipse, or a hyperbola.

If the coefficients of x² and y² are the same sign AND equal, the equation represents a circle.

If the coefficients of x² and y² are the same sign BUT different, you have an ellipse.

If the coefficients of x² and y² have different signs(one positive and one negative), then the graph will be a hyperbola.

I did this with my dad:

Now I'm exited to start calculus and advanced geometry.

Basic Algebra

11X + 23 = 4X - 5

That ↑ is called an equation.

My job is to find out the mystery number X.

First I get all the X's on one side.

So I subtract 4X from both sides of the equation.

11X + 23 - 4X = 4X - 5 - 4X

7X + 23 = -5

Next I subtract 23 from both sides.

7X + 23 - 23 = -5 - 23

7X = -28

Now I need to turn the 7 into an invisible 1.

So we divide both sides by 7.

7X / 7 = -28 / 7

X = -4


Let's check our answer.

11·-4 + 23 = 4·-4 - 5

-44 + 23 = -16 - 5

-21 = -21

Rate * Time = Distance

If you go 70mph for one hour....you'll go 70 miles.

If you go 70mph for two hours....you'll go 140 miles.

If you go 70mph for half an hour.... you'll go 35 miles.

The rule is:

rate * time = distance

In my algebra book sometimes they mix it up.

Sometimes they give the distance and the rate and have to solve for time.

Sometimes they give the distance and the time and have to solve for rate.

That is easy:

rate = distance / time

time = distance / rate

Another thing they want me to do is to find the average speed:

average speed = total distance / total time

The Number Line

I just finished chapter 1 in my algebra book.

I learned about sets, natural numbers, whole numbers, integers, and infinity.

Did you know that:

10 - (-41) = 51

???

Math!

Today I finished 6th grade math. That ↑ was my last book.

Tomorrow I am starting algebra!!!

Linear Equations

In algebra now I'm solving some equations with 2 variables.

For example:

x + y = 10

Here are some of the solutions to this equation:

(1,9)
(-3,13)
(9,1)
(1.5,8.5)
(13,-3)
(1000,-990)
(0,10)
(-33.2,43.2)
.
.
.

As you can see there are infinite solutions (x,y) to this equation. In fact, there are infinite solutions to any equation with an x and y.

The best way to describe these solutions is with a graph.

This graph is always a straight line that extends in both directions to infinity.

The easiest way to draw these lines is by computing the x and y intercepts.

All you have to do is set x equal to 0 and solve for y. Then set y equal to 0 and solve for x.

Then you have two points that you can connect to draw the infinite line!

For example, the intercepts of x + y = 10 are (0,10) and (10,0).

The other common way to graph a line is with slope-intercept form: y = mx + b where m is the slope of the line and b is the y-intercept.

More Algebra

Now I am studying Polynomials.

Here is an example:

6x² - 7x + 1

Here is another example:

10x⁵ + 4x⁴ - 13x³ + .5² - 102

Each part of a Polynomial is called a Monomial. My dad calls them "terms".

The number in front of each term is called a coefficient.

The little numbers above the x's are called exponents.

When x-terms have the same exponent they can be easily combined by adding their coefficients.

For example:

3x⁴ - 17x⁴ = -14x⁴

(-1/3)x⁵ + (1/2)x⁵ = (1/6)x⁵

Two terms or two monomials, make a "binomial". (Bi - means two).

You can also multiply, divide, and factor Polynomials