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Inconsistent and Dependent Systems of Equations - Problem 2

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

Some systems of equations will have no solution, meaning that the graphs of the lines never intersect. In other words, the lines are parallel. Remember that two parallel lines have the same slope but different y-intercepts. However, since equations of the same line could be written in different forms, it is not always easy to tell that two lines are parallel just by looking at the equations. When solving a system of equations using substitution or elimination, if the result is a statement that is not true -- for example, 0=3 or 2=-5 -- then the system has no solution. This means that there is no value of x and y that could make both equations true.

In this system of equations I'm going to look at this and think about using substitution because y is already isolated. Y is equal to the expression 3/2X take away 1. So I'm going to rewrite this first equation only instead of right there I'm going to write the expression 3/2x take away 1.

Here's what I mean 2 times now I'm using my substitution minus 3x equals 6. Okay let's go to through and solve that and see what we get.2 times 3/2 is equal to 3, oh-oh! This is funny because 3x and -3x when I go to combine like terms I get 0x. I'm getting -2 equals 6 that's not true -2 does not equal 6. That means that I have some special system of equations.

Go back to the very beginning of this problem with me. I'm going to erase this work and we'll go look at what we are given. I have two equations this one is already in y equals mx plus b form I'm going to rewrite this guy into y equals mx plus b form and we'll see if we see anything fishy.

Start by adding 3x to both sides so I'll have y equals 3x plus 6, divide everything by 2 so I have y equals 3/2x plus 3. That's my first equation rewritten, that's my second equation. Do you notice anything? They are parallel guys they have the same slope. We know parallel lines never cross. Never cross that means there's no solution. Never cross because parallel lines have the same slope, they are equally steep and that's how you can tell by looking at these two equations that they are parallel, and the other thing is that the solution to a system of inequalities is where the lines cross. We are saying parallel lines never cross therefore there is no solution.

So when you get something in your solving process like -2 equals 6 or 5 equals 100 something where you get something that's not true, what's going on there is you are working with 2 parallel lines. We call that no solutions and the other word for this that you might see in your textbook or classroom you is inconsistent.

Good luck with those guys I know you can do it just be really careful when you are looking at slopes especially watch out if you have parallel lines in your system of equations.

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