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Solving a System of Linear Equations in Two Variables - Problem 3

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Solving a system of linear equations. Whenever we solve a system of linear equations we first need to figure out if we want to do substitution or elimination. So for this problem behind me we do have a single x by itself, which means that substitution could be a pretty good option. However we also have a fraction on our y and in general most students don’t like dealing with fractions so what I want to do is get rid of that fraction and by getting rid of the fraction all we have to do is multiply that entire equation by its least common denominator or in this case, 2.

So by multiplying this entire equation by 2, we now have 2x plus 7y is equal to 4. Our coefficient on x is no longer 1 so it’s really not the best option for substitution which means we’re going to have to go to elimination.

Looking at our top equation, remember this middle equation I just rewrote in red because we multiplied by 2, we’re looking for getting a coefficient the same. So 2, 6, 7, 3, 2 and 6 are smaller number so let’s try to get rid of our x term.

To get rid of our x we want to turn this into a 6. I’m actually going to turn it into a -6 because I prefer to add. Basically what that means is I multiply this by -3 and I’m going to bring down our top equation so we can have them right next to each other. This turns into 6x plus 21y is equal to 12. That just came down to here and then multiply through by -3 and we end up with -6x minus 21y is equal to -12.

I chose the -3 so I would add my answers and not have to subtract; and when I add what happens is everything cancels out. Our 6s are equal and opposite and our 12s are equal and opposite leaving us with 0 is equal to 0. So what that is, is a true statement. Zero is in fact equal to zero so what means is these lines are actually going to be the same line.

Careful with how you write your notation, okay, your answer. A lot of people want to say infinite solutions, all reals, anything like that but it’s not the case. What actually is happening is any point on these lines is going to satisfy both lines but it's not going to be any point out there. I can’t just say a million zero and assume it's going to work on these lines. So what you have to do is you can say all solutions on the line, you could write your answer in set notation. There is a number of different ways of writing your answer but make sure you avoid just saying all reals or infinite solutions because you do have to be a little bit more specific than that.

There’s a number of different ways of writing it, check with your teacher to see which way they are going to want you to show, because I really don’t have time to go into all the different answers that could be out there.

Looking at a system of linear equations, getting rid of our fractions, elimination and then interpreting our answer zero is equal to zero. Zero is equal to zero tells us that we have parallel lines so there’s a number of different ways of writing that out but make sure you avoid the common mistake of saying infinite solutions.

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