Are the equations in slope-intercept form? What information can the slope-intercept form of an equation give you?
Number of Solutions: Infinitely many Graph:
Practice makes perfect
By graphing the given equations, we can determine the number of solutions to the system. This will be indicated by the point at which the lines intersect. To do this, we need the equations to be in slope-intercept form to help us identify the slope m and y-intercept b.
Writing the Slope-Intercept Form
Let's rewrite each of the equations in the system in slope-intercept form, highlighting the m and b values.
Given Equation
Slope-Intercept Form
Slope m
y-intercept b
y=4/3x-2
y= 4/3x+( -2)
4/3
(0, -2)
3y-4x=-6
y= 4/3x+( -2)
4/3
(0, -2)
When converted into slope-intercept form, we can see that these two equations are actually the same. This means that there are infinitely many solutions. We will demonstrate this with the graph below.
Graphing the System
To graph these equations, we will start by plotting their y-intercepts. Then, we will use the slope to determine another point that satisfies each equation, and connect the points with a line.
The equations overlap at every possible point. Since the equations overlap, they are also "intersecting" at every point along the line. Therefore, the system has infinitely many solutions.
