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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. 


Slope-Intercept FormPoint-Slope Form



$x$ and $y$ are the (x,y) coordinates from any point on the line.


$x_1$ and $y_1$ are the (x,y) coordinates from any point on the line.

$x_2$ and $y_2$ are the (x,y) coordinates from any other point on the line.

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:



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


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. 



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