Exercise 1 Page 363

An alternative method to determining the number of solutions to a system of equations by graphing is to compare the slope and y-intercept of the equations. y=mx+ b

To do this use the slope-intercept form of each equation, where m is the slope and the point (0, b) is the y-intercept. There are three possibilities when comparing two linear equations in a system.

Slope	y-intercept	Graph Description	Classification
m_1 ≠ m_2	irrelevant	intersecting lines	consistent, independent
m_1=m_2	b_1≠ 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-intercept b
y=2x-1	y=2x+( -1)	2	(0, -1)
y=-2x+3	y=-2x+ 3	-2	(0, 3)

Comparing the slopes, we see that they are not equal, so the lines are intersecting. This means that the system is consistent and independent.