If the lines have different slopes, then there is only one solution. If they have the same slope but different y-intercept, then there is no solution. Finally, if the lines have the same slope and the same y-intercept, then there are infinitely many solutions.
The system is consistent and independent.
Practice makes perfect
An alternative method for 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=-3x+1
y=-3x+( 1)
-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.
