How do you know how many solutions there are to a system of equations?

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• If both sides equal zero is that also a zero solution equation?

  • 
    

    If you are solving a system of equations and get to the point where you have: 0 = 0; this is an Identity (a situation that is always true). In a system of equations it means that the 2 equations are actually the same line (visualize one line sitting right on top of the other). So, the system has a solution set of all the possible points on that line.

    Contrast that with a solution like: 0 = 2. This is a contradiction (it is always false). This indicates that the 2 equations have no points in common. So, they would be parallel lines.

• y=−2x−4
  y=3x+3
  ​How many solutions does this system have? Pls help!

  • This system of linear equations have only one solution.
    That is because this system of equations is written in slope-intercept form:
    y=mx+b,
    In which m is the slope and b is the y-intercept.
    So in the first equation, -2 is the slope.
    And in the second equation, 3 is the slope.
    And it becomes very obvious -- two lines with a DIFFERENT slope will always intersect at some point!
    So this system has only one solution.
    P.S. sorry for answering 9 months later!

• I'm purposefully trying to make a difficult question.
  What is the solution to this system of linear equations?
  y=mx-1
  y=(m-1)x-2
  Where m = Graham's Number
  I figure it's something VERY close to (0,-2) because the two lines will almost seem to be on top of one another in an almost vertical line.
  To be a little more accurate, I know the solution will have an x value slightly more negative than 0 (-0.000...1) and a y value slightly more negative than -2 (-2.000...1).
  How do I show my solution algebraically?
  Another question: Picking an arbitrary location on the y-axis, how would I show the distance from those two lines?

  • I got a different answer to your first question.
    Through substitution, x = 3.
    So the solution to the system of equations y = mx - 1 and y = (m - 1)x - 2 is the ordered pair (3, y).
    To find y, we simplify again and see that:
    y = 3(Graham's Number) - 5
    So the lines will intersect at (3, y) where y is an extremely big number.

• I have difficulty understanding this kind of equations:
  ​y=4x−7
  ​y=x−3
  even when I get a hint, the hint is talking about another equation!

  • You have probably already figured this out by now, but you can use the second equation to solve for the first one :D Since the second equation says that y=x-3, you can substitute "x-3" into the first equation so that there's only one variable (x). Then you can solve the rest of the equation, I believe.

• How did you get -36 on the first problem

  • He multiplied or divided the whole second equation by -1 which in effect just changes the signs in the whole equation such that 5x - 9y = 36 becomes - 5x +9y - - 36

• On the last question, Sal wrote the co-ordinates 4/5, -1 but, how does he know that it' exactly one solution?

  • If you mean the one before the last exercise, try to think this way:

    There were 2 equations:

    5x-2y=6
    5x+3y=1

    Those 2 equations forms a system of equations right?

    To validate if the system has indeed only one solution, all of the lines within the system must have a different y-intercept.

    Since there is 2 equations in the system, we can say that there are 2 lines as well.

    To check their y-intercept you can assume x is zero for all of them.

    5(0)-2y=6 -> -2y=6 -> y=-3
    5(0)+3y=1 -> 3y=1 -> y=1/3

    Since their y-intercept is different, there is only 1 solution to the system.

    But, there are many ways to solve something like this, what Saul did was following the exercise clue: "Albus takes several correct steps that lead to the equation 5y=-5", which means that Albus joined both lines "behaviors" (equations) and ended up with a single equation (behavior).

    And if you observe carefully, you'll notice that what he did was a subtraction.

    5x-2y=6
    -
    5x+3y=1
    =
    -5y=5
    -5y(-1)=5(-1)
    5y=-5

    And that leads us to a different way of knowing if a system equation has one or more solutions, by solving them instead of analyzing its behaviors.

    If you join the behaviors (by adding, subtracting or isolating a variable and then merging the equations) and get a result different from 0=0 it means that the equation has only one possible solution (which will be either an 'y' or a 'x', that means where the lines encounter each other, in the case of the exercise Albus gave you the 'y').

• 5x - 9y = 16
  5x - 9y = 36
  Both 16 and 36 are equal to 5x - 9y. This means 16 = 36, but they aren't equal. I don't understand.

  • Which also means that the two lines created by these equations are parallel (they both have a slope of 5/9, but different y intercepts - -16/9 and -4).

• Sooo, this is my logic:
  after solving for y, I get an equation, say K.x +C(where K is coefficeint of x and C is a constant). Then, the equations have-
  1. No solutions if K is same but C is different in both equation.
  2. infinite solutions if K and C are equal for both equations.
  3. and, one solution if K is different.
  Is this right?

  • Yes, that is correct.
    K in this instance is called the 'slope' of a line. It's the number of steps that 'y' will grow at every step of 'x'. C is the y-intercept.
    When both equations are equal, you'll get infinite intersections, since the two lines overlap.
    When both equations have the same slope, but not the same y-intercept, they'll be parallel to each other and no intersections means no solutions.
    When both equations have different slopes than regardless of the y-intercept they'll intersect for certain, therefore it has exactly one solution.

• At

  2:07

  , even if the 0 had an x, like 0x = -20, it still wouldn't work, right? Because everything times 0 = 0, not -20.

  • Yes, in fact, x is undefined in that equation because to solve for it, you would have to divide by 0 which is undefined.

• How would you know if a system has exactly one solution, and i didn't get how in

  5:37

  (4/5, 1) represents exactly one solution

  • I'm not sure what you mean. That is the answer. You only have one place on the graph where the two lines intersect. Therefore, you only have one answer that satisfies both equations.

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