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# Worked example: equivalent systems of equations

Sal analyzes a couple of systems of equations and determines whether they have the same solution as a third given system.

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• I’m not getting the explainations, why are we sure that we get the same solution when we add the two equations and replace the second line with that ?
Why are we sure the new line is going to cross the first line at the exact same point ? •  We know the new line crosses the first at the same point because we know that point satisfies the new equation. The solution point is the point that satisfies both equations, that is, the x and y that make the two sides equal in both equations. So if you add things that are equal to things that are equal, the result must be equal, so the same solution must satisfy the new equation.

If you want to think of it algebraically, consider the two equations:
`ax + by = c` and `dx + ey = f`
When you add the left hand sides, you get `ax + by + dx + ey`, (you could combine like terms here, but it's easier to think of them separate). Assuming the system is solvable, we know there is some `x` and `y` that solve the system, so let's see what that same `x` and `y` do in the new expression. Since that `x` and `y` satisfy both of the original equations, the first part will evaluate to `c`, and the second part will evaluate to `f`, so the whole expression will evaluate to `c + f`. This means the same `x` and `y` will satisfy the new equation `ax + by + dx + ey = c + f` therefore line represented by that equation must pass through that point.
• In any linear equation, why does multiplying or dividing give the same line on the graph but adding or subtracting does not give the same line?
For example,
m+n=a+b
If we multiply by 2,
2(m + n )= 2(a +b)
If we plot on graph, both of these equations give the same line.
But if
m+n=a+b
Then, m+n -a =a+b-a
But these two equations have different lines • At , Sal said that the solution of the system is not going to change as long you as multiply both sides of the equation by a scaler. Would dividing both sides of the equation give the same result? • It doesn't make any sense to me . Sal subtracts one equation from the other and still, it gives the same solution when I graphed it in my notebook. Why does that make sense? • Adding one equation to th other or subtracting both work.

The way i think of it is pretty simple, I just hope it makes sense to you. Let's stat witht he two euations we start with in the video.

x + 2y = -1
-4x + 5y = 1

The point is to find x and y that make both true. Well if both are true we can actually get more creative. keep in mind, we are saying both are true.

Now, we could add 1 to both sides. so it would look like x + 2y + 1 = -1 + 1. So it's just like normal algebra here. Well, the second equation says 1 = -4x + 5y, and since we know it has to be true we can instead write one (or both) of those 1s as -4x + 5y. This would make x + 2y + 1 = -1 + 1 become x + 2y + (-4x + 5y) = -1 + 1 and then if you simplify everything you get -3x + 7y = 0. And this is the same as addng the two equations together.

Let me know if this doesn't make sense. Also, keep in mind you can combine this with multiplying an equation by a number. So x + 2y = -1 is also saying 2x + 4y = -2, which is 2 times x + 2y = -1.

Again, let me know if this doesn't make sense.
• If equivalent equations are equations that have the same solution , does that mean that the equations in a system of equations are equivalent equations? • The answer for that question is no! because they can also be not equivalent when the left sides match but the right sides do NOT match! See the next video for more info! The point of having equivalent equations is to make sure one of the students equation should have the left AND right sides match to the teachers equation! and then you can multiply the other equation because we are permitted! Good question N!
• Why would adding or subtracting 1 equation from another to get a different equation lead to the same solution? Would it be through simplification or something?

And would you get the same solution if you multiply or divide 1 equation by another to get your second equation in the system? • In the teacher's system if we add x and -4x, we will get -3x in the Vivek's second equation. No matter what "solution value" you give to the x in teacher's sysem, if you add two equations together (x - 4x) you will ALWAYS get -3 times that value you gave to x. The same is true for y values.

For instance: The solution to the teacher's equation system in this video is: (x= -7/13, y= -3/13)
When you add two equations and give the value -7/13 to x, the x part will be:
x-4x = -7/13+28/13 =21/13 which exactly equals -3x

If the x solution of the teacher's system was 1 million, nothing would change:
x-4x = 1 million - 4 million = -3 milion which exactly equals -3x

Also DON'T FORGET that when we add x and y's on the left side we also add the right sides of two equations: -1+1 = 0 in order to maintain equality.

And yes you can also multiply or divide as long as you do the same operation to the right side
• At , what is a legitimate operation? • Vivek's second equation isn't actually been multiplied by the teacher's second equation! so I wonder if Vivek's system of equation is NOT equivalent to the teacher's system of equation? why would it be equivalent?   