1. Give an example of a system of three linear equations with two variables that
ID: 3103385 • Letter: 1
Question
1. Give an example of a system of three linear equations with two variables that has exactly one solution.2. Give an example of two linear equations with three variables that has no solutions.
3. Find a number b such that the system of linear equations:
3x-2y=1
4x+by=5
has no solutions.
Explanation / Answer
1. The idea here is to think about when 3 linear equations have exactly one solution. Well, this is equivalent to asking construct three lines (linear equations: y = mx+b) that intersect at only point! So consider something like x+y = 1, x-y = 1, and 3x + 3y = 3. (How did I get these equations? I simply drew 2 lines that intersected at only one place and found the equations for them and then took the scalar multiple of one of the equations (which gives me the same line back). I made them extremely simple so the equations would be nice). Solving this system we see that x = 1 - y. So then, x - y = 1 --> (1-y) - y = 1--> -2y = 0 --> y = 0. ==> x = 1. We check 3(1) + 3(0) = 3. All three equations are solved. We note this is THE ONLY solution because 3x+3y = 3 is equal to x-y=1 so it's essentially the same line (graph it if you want to double check). 2. The motivation for our solution is to think of 2 planes (equation w/ 3 variables) that do not intersect. So consider x+y+z = 1 and x+y+z = 2. Obviously, there are no numbers x,y,z that when you add them give you 1 and then when you add them again give you 2. Thus these 2 equations have no solutions. 3. We use the idea as #2 but in 2 dimensions. Well we note that if we multiply 3x-2y by some constant but change the value it's equal to (not by the same factor we multiplied by), then there can be no solutions because adding a 2 numbers up (the x and y) will give you a unique value [e.g. see above]. Well, we notice 3x went to 4x. Or alternatively, 3x * 4/3 = 4x. Lets try multiplying 3x -2y = 1 by (4/3). We get 4x - 8/3 y = 4/3. So if b = -8/3, then this equation will have no solutions because if it equaled 8/3, then 4x - 8/3y would HAVE to equal 4/3! but we are saying it equals 5, therefore there can be no x,y that satisfy these two equations when b = -8/3.
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