GRAPHING LINES

In this section, we firm up the relationship between a line in the coordinate plane
and its description as an equation in two variables.

In the process, some general strategies for graphing a line are discussed.

DEFINITION linear equation in two variables
A linear equation in two variables ($\,x\,$ and $\,y\,$) is an equation of the form: $$ ax + by + c = 0 $$ In this equation, $\,a\,$, $\,b\,$, and $\,c\,$ are real numbers.
The numbers $\,a\,$ and $\,b\,$ cannot both equal zero.

Every linear equation in two variables graphs as a line in the coordinate plane.
Every line in the coordinate plane has a description as a linear equation in two variables.

The equation $\,ax + by + c = 0\,$ is often called the standard form or general form of a line.
IMPORTANT THINGS TO KNOW ABOUT LINEAR EQUATIONS IN TWO VARIABLES
SLOPE-INTERCEPT FORM OF A LINE, $\,y = mx + b$
Every equation of the form $\,y = mx + b\,$ graphs as a non-vertical line.
The slope of the line is $\,m\,$ (the coefficient of the $\,x\,$ term).
The line crosses the $\,y$-axis at the point $\,(0,b)\,$.
Since the equation $\,y = mx + b\,$ so clearly displays the slope and $\,y$-intercept,
it is called slope-intercept form.
IMPORTANT THINGS TO KNOW ABOUT SLOPE-INTERCEPT FORM
EXAMPLES:
Question:
Consider the line $\,2x - 3y + 5 = 0\,$.
Write the equation in the form $\,y = mx + b\,$.
What is the slope of the line?
What is the $\,y$-intercept?
If you start at any point on the line, how could you move to get to another point?
Solution:
To put the equation in $\,y = mx + b\,$ form, solve for $\,y\,$:
$2x - 3y + 5 = 0$(original equation)
$-3y + 5 = -2x$(subtract $\,2x\,$ from both sides)
$-3y = -2x - 5$(subtract $\,5\,$ from both sides)
$\displaystyle y = \frac{-2x - 5}{-3}$(divide both sides by $\,-3\,$)
$\displaystyle y = \frac23x + \frac53$(write in the most conventional way)

slope:   $\displaystyle \,m = \frac 23 = \frac{\text{rise}}{\text{run}}$

$y$-intercept:   $\displaystyle b = \frac53$

To get to a new point, you could move up $\,2\,$ and to the right $\,3\,$.
(There are, of course, other correct answers.)
Question:
Consider the line $\,2x - 3y + 5 = 0\,$.
What is the $\,x$-intercept? (Give the coordinates.)
What is the $\,y$-intercept? (Give the coordinates.)
Solution:
To find the $\,x$-intercept, set $\,y = 0\,$ and solve for $\,x\,$:
$2x - 3y + 5 = 0\,$(original equation)
$2x - 3(0) + 5 = 0\,$(set $\,y = 0\,$)
$2x = -5\,$(subtract $\,5\,$ from both sides)
$\displaystyle x = -\frac52\,$(divide both sides by $\,2\,$)
The $\,x$-intercept is $\,(-\frac52,0)\,$.

To find the $\,y$-intercept, set $\,x = 0\,$ and solve for $\,y\,$:
$2x - 3y + 5 = 0\,$(original equation)
$2(0) - 3y + 5 = 0\,$(set $\,x = 0\,$)
$-3y = -5\,$(subtract $\,5\,$ from both sides)
$\displaystyle y = \frac53\,$(divide both sides by $\,-3\,$)
The $\,y$-intercept is $\,(0,\frac53)\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Finding Equations of Lines
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
Answers are reported as fractions in simplest form.
PROBLEM TYPES:
1 2 3
AVAILABLE MASTERED IN PROGRESS