How can we use consistent and dependent systems?
A system of equations is considered consistent when it has at least one solution. Therefore intersecting lines and coinciding lines are both consistent systems. We can get more specific depending on whether they are intersecting or coinciding.
A consistent system with one solution is known as an independent system. The graph of this system would result in intersecting lines that cross at one point. If you are given a system and put the equations into slope-intercept form, the slopes must be different.
A consistent system with one solution is known as an independent system. The graph of this system would result in intersecting lines that cross at one point. If you are given a system and put the equations into slope-intercept form, the slopes must be different.
A consistent system with an infinite number of solutions, it is dependent. These are coinciding lines or lines that are exactly the same. The graph of this system would look like a single line. If you are given a system of equations and put the equations into slope-intercept form, the two equations would look exactly the same. Additionally, if the equations were put in standard form, one of the equations would be a multiple of the other.
If the system of equations has no solution, or are parallel lines, then it is said to be inconsistent. These are parallel lines. The graph of this system would have two lines that never intersect and therefore have no solution. If you are given a system and put the equations in slope intercept form, the slope would be exactly the same, but the y-intercepts have to be different. If you are given the equations in standard form, the Ax + By would be multiples of each other, but the C’s would be different.
Sample Math Problems
Question:
Classify the following graph as consistent independent, consistent dependent, or inconsistent.
Solution:
Since the lines are parallel, there are no solutions. Therefore the system is considered inconsistent.
Question:
Classify the following system as consistent independent, consistent dependent, or inconsistent
2x + 4y = 20
x + 2y = 5
Solution:
If you are to put these equations into slope-intercept form you would get the following:
If you are to put these equations into slope-intercept form you would get the following:
2x + 4y = 20 → y =
x + 2y = 5 → y =
Based upon the slope intercepts form, they have the exact same slopes but different y-intercepts so the lines are parallel. This system is classified as inconsistent.
We could have also used standard form and saw that when you multiplied the bottom equation by two, the result would be 2x + 4y = 10. The Ax + By is the same as the top equation, but the C is different.
Question:
Classify the following system as consistent independent, consistent dependent, or inconsistent
21y - 4x = 14
12x - 3y = 22
Solution:
If you are to put these equations into slope-intercept form, you would get the following.:
If you are to put these equations into slope-intercept form, you would get the following.:
21y - 4x = 14 → y =
Question:
Classify the following system as consistent independent, consistent dependent, or inconsistent
56x - 2y = 12
28x - y = 6
Solution:
If you are leaving this in standard form, you can see that the second equation can be multiplied by 2 and will result in the first equation. Based upon this, these two equations, when put into slope-intercept form, would simplify to the same equation. These would create coinciding lines and be classified as consistent and dependent.
