Linear Equations
"y = mx + b"
In this video, a line teaches students about the properties of lines and linear equations. The song describes various properties of lines. They don’t curve and only cross the x-axis and y-axis once each. The song also explains linear equations, x-intercept and y-intercept, slope, and how to find the slope-intercept form of a linear equation.
I can go on forever. I am the shortest distance between two points. And I can relate to x and y. Who am I?
I’m not lying, I’m a line, you don’t need to ask me,
I never curve when these nerds try to graph me.
Pick a point, anywhere where I lie,
You’ll find a value for x and find a value for y.
Then slide, go for a ride,
A new value for x, a new value for y.
Check how my x gets negative,
And my y gets negative, I’m the best who ever lived.
On the graph, there’s two lines I’m crossing,
I only cross each once, no not often.
They ask, “What’s your y-intercept?”
That’s where I cross the y-axis, take it from a vet.
They ask, “What’s your x-intercept?”
That’s where I cross the x-axis, better get in step.
And line up, you know we make it pop,
Lines are forever 'cause we never ever stop!
When y = mx + b,
b is the y-intercept, you’ll see.
m is the slope, rise over run.
They’re waiting till we stop,
But that day will never come. (x2)
You can write me as a linear equation,
A line equation that describes my situation.
I have values for x and y that change,
So write x and y on the page.
Other numbers are how x and y relate,
x might be two times as big plus 8.
You can do operations on either side, yes,
But if you do it to the right, do it to the left.
Get y isolated, that’s wise and fun,
You’ll be able to see how I rise and run.
You’ll have y = mx + b,
Is this slope intercept form? It must be.
b is my y-intercept, boss,
Where me and the y-axis cross.
And m is my slope, rise over run,
How fast I go up over how fast I go run to the right.
I know that might sound crazy,
But get it straight like a line, not wavy.
I give praise to the ruler that made me,
One dimensional, but you can’t outrace me.
When y = mx + b,
b is the y-intercept, you’ll see.
m is the slope, rise over run.
They’re waiting till we stop,
But that day will come. (x2)
A line is a straight series of points in a particular plane that extends infinitely in both directions. Lines are straight; they don't curve. So they are the shortest distance between two points. Lines are infinitely long, so you can't measure the length of a line in units. Lines are also one-dimensional, which means that don't have any width at all. For this reason, you've never seen lines in real life, only representations of them. Lines are important mathematical ideas.
You can find the slope of a line by using any two points on that line. A line is made up of an infinite series of points, and each one has an x coordinate and a y coordinate.
A line only crosses the x-axis once and the y-axis once. The only exceptions are lines that run parallel to the x or y axis. Two parallel lines never cross.
The y-intercept is the point on the y-axis where the line crosses it. It's where the line and the axis "intercept" each other.
The x-intercept is the point on the x-axis where the line crosses it.
The relationship between x and y is the same at any point on the line. That means that no matter where you move on the line, x will always be the same amount bigger or smaller than y.
The relationship between x and y is the same at any point on the line. That means that no matter where you move on the line, x will always be the same amount bigger or smaller than y.
If you see the equation written in a different format, you'll probably want to move things around until you have y by itself. In other words, solve for y. This will put it in a format where all of its key features are easy to see.
So if x is two times as big as y, plus 8...
x = 2y + 8
2y = x – 8
y = 1⁄2x – 4
The equation y = mx + b is called the slope-intercept form because the slope (m) and y-intercept (b) of the line are right there in the equation. This is the easiest form to use when graphing a line, or to write the linear equation when given points on the line.
In the equation we have now, what is the slope? What is the y-intercept?
y = 1⁄2x – 4
Slope, or m, is calculated by measuring the rate at which a line goes up versus the rate at which it goes to the right. ‘Rise over run’ summarizes it nicely--you put the amount that a line has risen on the y-axis over the amount that the line runs to the side on the x-axis at the same time.
In the equation we have, y = 1⁄4x – 4, the slope is 1⁄4. That means that if a point is 1 higher on the y-axis than another point on the line, it is also 4 further to the right.
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