Solving for Variables Using Graphs/Coordinates

Remember, every linear equation forms a line.  Each line contains a set of points whose coordinates are given as $(x, y)$.  The coordinates $(x,y)$ not only show the points on the line, but they provide the $x$ and $y$ variables in the linear equations (e.g., in $y = mx + b$).

If you are given a coordinate (or both coordinates) of a point on a line, you can plug those coordinates into the x and y of the linear equation. 

Coordinates are written as $(x,y)$.  $x$ is the coordinate on the horizontal axis; $y$ is the coordinate on the vertical coordinate.  You can write lines in slope intercept form, $y=mx+b$, or point-slope form, $(y_1-y_2)= m(x_1-x_2)$. Simplified, these equations, they are the same!  These equations work with the $(x,y)$ coordinates from ANY point on the line.

Whether you are working with slope-intercept form or point-slope form, you will need to have the slope of the line ($m$).  

You may be given $m$, or you may have to find it:

You can use $\dfrac{\text{rise}}{\text{run}}$ to find slope from a graph.

Or you can use the slope formula $\dfrac{y_2-y_1}{x_2-x_1}$ to find the slope using two points.

When you are given points on a line, you can always plug the $(x,y)$ into either a slope-intercept, or a point-slope equation.  Either method will yield $b$ and the equation of the line.

If you are just given two points, and one is the y-intercept, you can use point-slope form, but you can also just put the y value of the y-intercept into $y=mx+b$ as $b$ for a faster result. 

These two equations give you all of the tools you need to determine the equation of almost any line.

Example: The coordinates of Point A are (3,$a$). If Point A lies on the line with the equation $y-2=4(x-5)$, what is $a$?

First, figure out what you know.

The equation of the line is:  $y-2=4(x-5)$

Another point on the line is: (3,$a$)

They are using: point-slope form

The slope of the line is: 4 (look where the $m$ is in point-slop form -- the given equation has a 4 in that position).

Next, figure out how you can add your additional information to the given equation.

You have another point: (3, $a$)

Coordinates are listed (x, y), so:

$x=3$

$y=a$

Plug those additional values into the equation for $x$ and $v$:

$$\eqalign{y-2&=4(x-5)\\a-2&=4(3-5)\\a-2&=4(-2)\\a-2&=-8\\+2&\;\;+2\\a&=-6}$$

When finding equations of lines, or solving for variables in equations of lines, it's critical to remember that the coordinates of every point on a line can be used in the equation for that line.  Plug in $x$ and $y$ and you should be able to find the variable you need.