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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'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?
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
• Why am I getting Korean subs?
• That happens to me sometimes as well. It i easy to change, just click the CC to turn them off or if you want English subtitles go to the gear and change to English. Or you could starting practicing Korean! :)
• 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.
• So, is this how it works?

x=x is always infinite solutions
Ax=B is always one solution
A=B is always no solution

(A and B can be any number but they have to be different for my example/question)
• Yes, that's a good summary. But be aware that in your 1st line (x=x), instead of x=x you could get a number = itself.