determine whether the sysyem is consistent, inconsistent, or dependent.
3x + 2y = 15
6x + 4y = 30
notice that if you multiply the fist equation by 2 you get exactly the second one.
What do your notes or text say about this situation?
To determine whether the system of equations is consistent, inconsistent, or dependent, we need to analyze the slopes and intercepts of the two equations.
Let's start by rewriting the given system of equations:
Equation 1: 3x + 2y = 15
Equation 2: 6x + 4y = 30
To find the slopes of both equations, we need to rewrite them in slope-intercept form (y = mx + b), where m represents the slope:
Slope-intercept form of Equation 1:
3x + 2y = 15
2y = -3x + 15
y = (-3/2)x + 15/2
Slope-intercept form of Equation 2:
6x + 4y = 30
4y = -6x + 30
y = (-6/4)x + 30/4
y = (-3/2)x + 15/2
By comparing the equations, we can see that they have the exact same slope (-3/2).
Now, let's analyze the intercepts:
Equation 1:
y-intercept = 15/2
Equation 2:
y-intercept = 15/2
Both equations have the same y-intercept.
Based on this analysis, we can conclude that the given system of equations is dependent since the two equations represent the same line. This means that there are infinitely many possible solutions.