The system of equations above has solution X, Y. Graphing is probably the easiest, safest and fastest way to answer this question, but solving gave us the same answer. If the lines are parallel (never crossing) they have NO solutions. Since they are two lines, they CANNOT have two solutions-two lines can cross one time or can coincide (be duplicates of the same line) in which they have the same solutions at an infinity of places. He quickly graphed the lines to show that they intersect at (0, 0) which is the same solution we found be solving.
Sal decided to use the fact that this is a system of linear equations, which means it represents two lines. So there is one solution and it also explains why y can equal 9y. Plug that into your original equation to find out that when y = 0, x = 0
Instead you need to use algebra to isolate the y by subtracting y from both sides: You might be tempted to say IMPOSSIBLE! y cannot equal 9y If you substitute the value for x into the equation for y you get y = 3 (3y) If you try substitution to solve this system, you get some strange equations. If you are able to get any solution, you CAN say that the "zero solutions option" is not correctĪnother reason it would be tricky to solve is that this problem is tricky to solve. The answer is whether there are two solutions, only one solution, no solutions or infinite solutions. The answer is not the number that you would get when you solved the equations. In §7.5, he attempts to address the confusion this caused one student who complained, but I would think it better to bother to use set-builder notation and thus to avoid the problem altogether, as do three of the four freely-available linear algebra textbooks recommended by the American Institute of Mathematics (the one that doesn’t omits null spaces entirely).ĭr Grinfeld, and perhaps his students, likely would make fewer mistakes in Gaussian elimination were he to use augmented matrix notation, as is used in the said three textbooks.Yes, but you have to be careful. §2.12, “Linear Subspaces of the Plane”, is confusing enough that I think it ought to be redone - for now, read the comment under the video.ĭr Grinfeld habitually abuses notation for null spaces, writing a linear combination with unknown coefficients to represent the set of all linear coefficients of its form.
I think Dr Grinfeld's linear algebra material is certainly good for its price ($0.00), although my notes from going through the first eight or so chapters note the following small number of issues: There is a website named Lemma run by one Pavel Grinfeld that has a course in linear algebra that is similar to Khan Academy's more developed material, in that it has short lecture videos interspersed with exercises that are graded automatically by the website.
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