Exercise 1 Page 338

An alternative method for determining the number of solutions to a system of equations by graphing is to compare the slope and $y\text{-}$intercept of the equations. To do this, use the slope-intercept form of each equation, where $m$ is the slope and the point $(0,b)$ is the $y\text{-}$intercept. There are three possibilities when comparing two linear equations in a system.

Slope	$y\text{-}$intercept	Graph Description	Classification
$m_1\ne m_2$	Irrelevant	Intersecting lines	Consistent, independent
$m_1=m_2$	$b_1\ne b_2$	Parallel lines	Inconsistent
$m_1=m_2$	$b_1=b_2$	Same line	Consistent, dependent

Let's write the equations in the given system in slope-intercept form, highlighting the $m$ and $b$ values.

Given Equation	Slope-Intercept Form	Slope $m$	$y\text{-}$intercept $b$
$y=\text{-}3x+1$	$y=\text{-}3x+(1)$	$\text{-}3$	$(0,1)$
$y=3x+1$	$y=3x+(1)$	$3$	$(0,1)$

By comparing the slopes, we can see that they are not equal, so the lines intersect. The system is consistent and independent.