What do we notice about the equations of lines that intersect at one point? How about the ones that never intersect? And the ones that overlap entirely?
To do this, we will 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
Number of Solutions
m_1≠ m_2
irrelevant
intersecting lines
one solution
m_1=m_2
b_1≠ b_2
parallel lines
no solution
m_1=m_2
b_1=b_2
same line
infinitely many
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
8x+4y=12
y= -2x+( 3)
-2
(0, 3)
3y=-6x-15
y= -2x+( -5)
-2
(0, -5)
Comparing the slopes, we see that they are equal, so the lines are either parallel or the same. Looking at the y-intercepts, we can tell the lines are different because the point at which each line crosses the y-axis is different. This means that the lines are parallel and the system has no solution.
