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# Number of solutions to a system of equations algebraically

Sal solves several examples where he reasons about the number of solutions of systems of equations using algebraic reasoning.

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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? • 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 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! • 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? • How did you get -36 on the first problem • 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').
• At , even if the 0 had an x, like 0x = -20, it still wouldn't work, right? Because everything times 0 = 0, not -20. • 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. • 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. 