If the lines have different slopes, then there is only one solution. If they have the same slope but different y-intercepts, then there is no solution. Finally, if the lines have the same slope and the same y-intercept, then there are infinitely many solutions.
One solution
Practice makes perfect
An alternative method for determining the number of solutions to a system of equations is to compare the slope and y-intercept of the equations.
y= mx+ bTo 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
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 rewrite 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
3x+2y=1
y= -3/2x+ 1/2
-3/2
(0, 1/2)
4y=6x+2
y= 3/2x+ 1/2
3/2
(0, 1/2)
From the table we see that the slopes of the equations are not equal, so the lines must intersect. Therefore, the system has one solution.
