audio read-through Graphing Lines

Want some practice with preliminary concepts first?

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:

$$ \cssId{s11}{ax + by + c = 0} $$

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 $\,(0,\frac53)\,.$

Concept Practice

Answers are reported as fractions in simplest form.