Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 1. In this nine-minute video, Mr. Patrick shows what a homogeneous system of linear equations is, and shows what it means to have only trivial solutions. In the next vid...eo, the instructor works out an example that has non-trivial solutions.[more]

Homogeneous Systems of Linear Equations - Trivial and Non-Trivial Solutions, Part 2. In this ten-minute video, the instructor shows how to find solutions to a homogeneous system of linear equations that has non-trivial solutions.

This rap is from Rhythm Rhyme Results. It is audio only, lyrics and images are NOT included on the screen. (02:52)
This song teaches the elements needed to graph functions, including the quadrants on a coordinate plane, the equation of a line, an...d how to find the slope and intercept of a line. "Rise Up (Run Out)" then includes a real-life example explaining how to solve problems using a graph and the cartesian coordinate system. The teaching materials at the bottom of the page, including lesson plans, worksheets, activities, and games help teachers and homeschool parents generate teaching ideas for the song topic. Students will have a greater understanding of graphs and will be more effective at graphing linear equations.
Lyrics include:
Chorus
Rise up, run out, we’re mappin’ graphs to the RRR sound
We’re talkin’ points and slopes, axes and origins
y-intercepts, and quadrants, all four of them
(×2)
Verse I
The four quadrants start from the top right
Counter-clockwise from the positive x and y
You go around, hit the II and then III
Bottom left, negative, remember that please
Now if you’ve got a single point on the x-y grid
There’s a single pair of values telling you where it is
The x coordinate goes side to side
The y follows the comma to define the point’s height
A straight line between two points is a relation
You can represent it with a linear equation
y = m times x + b
If you wanna know the variables listen to me
m is the slope, that’s rise over run
It tells you how steep the line is gonna become
b is the intercept, that’s how you assess
The point at which the line goes through the y-axis
Chorus
Verse II
Let’s say you’ve got a job and you’re earning your pay
You start with 50 bucks and get 9 every day
If you wanna know over a year how much you’ll get
Calculating that graph might be your best bet
On the x, you’re measuring one day at a time
And your money gets tracked on the vertical y
Now you got some, and you haven’t even worked yet
50 bucks, put it up, that’s your y-intercept
After day one you got 59
Now you got two points, you can start drawing a line
The slope of that line, remember rise over run
It’s dollars over days and you got 9∕1
Remember b, I think that’s where we started at
50 dollars up on your graph, the y-intercept
And you know y = mx + b
So now you got y = 9x + 50[more]

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