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.
Infinitely many solutions
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
y=5x-1
y= 5x+( - 1)
5
(0, - 1)
10x=2y+2
y= 5x+( - 1)
5
(0, - 1)
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 the same because the point at which each line crosses the y-axis is the same. Therefore, the system has infinitely many solutions.
