Coordinates and Graphs of Lines

Solutions to Equations in Two Variables

Example

Consider the equation

2x + y  =  10

We see that when

x = 3      and      y = 4

we have

2(3) + 4  =  10

is true.  We say that (3,4) is a solution to the equation.  We call (3,4) and ordered pair.  The first number represents the x value and the second number represents the y value.  Notice that the ordered pair (1,2) does not satisfy the equation since

2(1) + 2  =  4

which is not 10.

Exercise

Which ordered pairs satisfy the equation

4x + 3y  =  24

1. (3,5)

2. (6,0)

3. (3,4)

T Tables

To find ordered pairs that satisfy the equation, we first solve (if possible) for y in terms of x.  Then we construct a T Table that consists of two columns.  The first represents the x coordinate and the second column represents the y coordinate.

Example

Given that

2x - y  =  4

we solve for y:

-y  =  -2x + 4                        Subtract 2x from both sides

y  =  2x - 4                            Multiply by -1

Now we construct a table by plugging in several values of x and find y:

x = 0:   y = 2(0) - 4  =  -4

x = 1:   y = 2(1) - 4  =  -2

x = 2:   y = 2(2) - 4  =  0

x = 3:   y = 2(3) - 4  =  2

x = -1:  y = 2(-1) - 4  =  -6

We create our table:
 x y 0 - 4 1 - 2 2 0 3 2 -1 - 6

This is called a T-table.

Exercise

Construct a T-Table for

2y - 6x = 8

The Coordinate Axes and Graphing

Just as we draw a number line to represent real numbers, we can also represent ordered pairs.  We draw two number lines that intersect each other at right angles.  We call the plane that the two number lines lie in the xy-plane. We call the horizontal line the x-axis and the vertical line the y-axis.  We let the right of the x-axis represent positive x values, while the top of the y-axis represents positive y values.  The intersection point is where x = 0 and y = 0 is called the origin.  The top right part of the plane is called Quadrant I.  The top left part of the plane is called Quadrant II.   The bottom left part of the plane is called Quadrant III.   The bottom right part of the plane is called Quadrant IV.  To represent an ordered pair (x,y) on the xy-plane, we walk x units to the right and y units upward.  Below are the points (1,2), (-1,1), (-2,-1), and (2,-1) plotted on the xy-plane.

 Definition The graph of an equation the collection of points (a,b) on the xy-plane such that (a,b) is a solution to the equation.

 Theorem An equation of the form            Ax + By = C   where A and B are not both 0, is a line.  Hence to graph an equation of this form, we need only plot two points.

Remark:  We usually plot the point where x = 0 (called the y-intercept) and the point where y = 0 (called the x-intercept.)

Example:

Graph

2x - 3y = 6

Solution:

We have the T-Table with the two points.

 x y 0 -2 3 0

We see that the x-intercept is at (3,0) and the y-intercept is at (0,-2).  Next we plot these points on the xy-plane and connect the dots.

Horizontal and Vertical Lines

Horizontal and vertical lines have special equation that we describe below.

 Definition An equation of the form             y = a is a horizontal line a units up from the x-axis An equation of the form           x = a is a vertical line a units to the right of the y-axis.