Manipulating line equations

Review

Last class topic was about how linear equations solutions can be thought as all points that creates a line. Whe have something like this:

\begin{equation*} y=3\cdot x +2 \end{equation*}

or

\begin{equation*} 2y+x=3 \end{equation*}

And we can choose wathever value we like for one variable, and the other gets fixed and related. There is one $y$ value for each $x$ value, and vice-veresa, buth this pairs are unique.

Getting the line

If we plot linear equations solutions, we'll get lines. That's easy. But how can one identify which equation formed a given line?

Slope



Play with $a$ value ¿what changes in the plot?

Intercept

Now, we fixed $a$ (the number in front of $x$), and let $b$ take different values ¿What does $b$ changes in the plot?



Come together

Now if we add both ideas, that $a$ it's about plot tilting (it points upwwards or dowwards) and $b$ stands for the $y$ value where the line intercepts the $y$ axis, namely, if $x=0$ then $y=b$. Any linear equation can be writen as:

\begin{equation*} y=a \cdot x + b \end{equation*}

There is nothing special about solving for y, it's just a convention of math people trhough history. We call $y$ the vertical axis, and $x$$ the horizontal. But it's pretty usual to solve for $x$ or to use other letters, because they are just placeholders, letters, and have no special meaning.

Run and Rise

A way of telling which function (equation) belongs to a given plot it's to find $a$ and $b$. Fin $b$ it's straight because one has to look at $y$ axis interception. But ¿what about $a$?. A method is as follows:

  1. Step on a point that belongs to the line (a solution, an $x$,$y$ pair)

  2. Run to the right (left will be ok too, you'll se later) any given units you want (1,2,3)

  3. Write down this value. We call it \(\Delta x\)

  4. Rise (or go down) until you reach the line again

  5. Count the units that took you there

  6. Write down this value. We call it \(\Delta y\)

  7. Do \(\frac{\Delta y}{\Delta x}\)

  8. Result is $a$

Test what you've leanred

Finding parameters

  1. Write a linear equation with $a=2$ and $b=-1$ .Plot it

  2. What's $b$ value in \(y=\frac{2}{3}x+2\)

  3. What's $a$ value in \(y=\frac{1}{3}x+2\)

  4. What's $b$ value in \(y+2=3x+1\)

  5. What's $a$ value in \(2y=\frac{2}{3}x+2\)

From plot to equation

Write the equation for each plot