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Inconsistent and Dependent Systems of Equations - Problem 1
When the equations in a system of equations represent the same line, the system will have an infinite number of solutions since the lines are overlapping at each point of the lines and not just at one point. Equations representing the same line might be written in different forms so it's not always easy to tell immediately that the equations are for the same line. When solving the system of equations using elimination or substitution, if the result is a statement such as 0=0 or x=x or 5=5, then this means that the two equations represented the same line and there are an infinite number of solutions. Any coordinate points plugged into the equations will make the equation true.
Here I have a system of equations that I'm asked to solve and they didn't specify what method so I get to choose. When I looked at this problem, the method that came to my head was substitution, because y is already isolated y is equal to the quantity 3x take away 1.
So what I'm going to do is rewrite the second equation only instead of where I see y I'm going to write 3x take away 1. Here we go 3x take away 1. Make sure to use parenthesis so you remember that negative sign is going to get distributed over the both of those terms.
Okay let's go through and solve for x 3x take away 3x plus 1 minus, oh-oh! I can tell something funny is going on here. 3x take away 3x is 0 Xs, 1 take way 1 is 0. What I have here is 0 equals 0 that's always true that's something funny.
Let me think about what's going on here. I'm going to erase what I've done let's go back and look at the original equations more carefully. This first equation is already solved for y let's try taking the second equation and solving for y, by adding y to both sides.
I'll have 3x minus 1 is equal to y. Do you guys see what's interesting about this equation and that equation? They are the same thing its exact sane line this is kind of like a trick problem.
They gave me the exact same line written in two different forms. The word we use to describe that is a dependent system, sometimes your textbook or your teacher will ask you to write the word that describes that this is a dependent system because its the same line. And the other thing that's interesting is I'm going to have infinite solutions. Any x-y pair that I want to like I could plug into both lines and I will get an inequality statement. I could plug in (5,4) I could plug in (1/3,195).
I could plug in 2, negative cloud it doesn't matter I can use any numbers I want to substituting into both lines and it will always work because it's the same line written in two different forms.
Any time you get something that looked like 0 equals 0 or 5 equals 5, any kind of statement like that that's always true, you are going to have a dependent system with infinite solutions.