Super fast systems of equations
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Solving Linear Systems by Graphing
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Graphing systems of equations
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Solving Linear Systems by Substitution
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Systems of equations with substitution
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Solving Systems of Equations by Elimination
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Systems of equations with simple elimination
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Solving Systems of Equations by Multiplication
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Systems of equations with elimination
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Systems of equations
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Special Types of Linear Systems
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Solutions to systems of equations
Special Types of Linear Systems Special Types of Linear Systems
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- Let's do some more systems of equations problem.
- In this video, we're going to encounter systems that might
- have no solutions, or that might have an infinite many
- solutions, and we'll label them with words.
- So let's start with one.
- Let's say we have 3x minus 4y is equal to 13.
- Let's say my other equation in my system is y is equal to
- negative 3x minus y.
- So the first thing-- this is kind of in a strange form
- right here.
- I want to get into the standard form and maybe I'll
- do elimination for this systems. Let me rewrite the
- top equation.
- We have 3x minus 4y is equal to 13.
- And let me rearrange this bottom equation here.
- So if I were to add 2y-- well, let me subtract y from both
- sides of this equation.
- So if I subtract y from both sides of this equation, it
- becomes 0 is equal to negative 3x minus 2y, or negative 3x
- minus 2y is equal to 0.
- So let me write that over here.
- And it looks nice because I have a negative 3x here, I
- have a positive 3x here.
- Looks well suited for elimination.
- So, I have negative 3x minus 2y is equal to 0.
- So let's add the left-hand side of this equation to the
- left-hand side of the yellow equation.
- And we're going to add 0 to the right-hand side of the
- yellow equation and we're essentially
- adding 0 to both sides.
- We're adding the same quantity to both sides, which we can
- always do with an equation.
- So, the left hant-side, the 3x cancels out with a 3x and
- we're left with negative 4y minus 2y.
- You get negative 6y is equal to 13.
- Divide both sides by negative 6.
- We are left with y is equal to negative 13/6.
- Now let's solve for x.
- And we can solve for x using either of these equations.
- Let's use that top one, just for fun.
- So we have 3 times x minus 4 times negative
- 13/6 is equal to 13.
- Now we have a negative times a negative, so those are both
- going to become positives.
- And then the 4/6, that's the same thing as 2/3.
- So this becomes 3x plus 2 times 13 which is 26/3, is
- equal to 13.
- Instead of 13, since I'm about to subtract 26/3 from both
- sides, let me rewrite 13 as 39/3.
- right?
- 39 divided by 3 is 13.
- So, let me subtract 26/3 from both sides.
- The left-hand side becomes 3x-- these cancel out-- is
- equal to 39 minus 26 is 13/3.
- And then we're going to want to divide both sides by 3.
- Or you can view it as multiplying both sides by 1/3.
- The left-hand side, we're just left with an x.
- The right-hand side, x is going to be equal to 13/9.
- So this system had a well defined solution.
- The solution is x is equal to 13/9 and y is equal to
- negative 13/6.
- It only has one solution.
- So if you think of these as lines, these two lines
- intersect in exactly one point.
- And a system like this, where it has exactly one solution,
- is called a consistent system of equations.
- And everything we've been doing so far has been
- consistent systems. Let's see if we can stumble upon
- something that's maybe a little less consistent.
- Let's say we have a system.
- Let's say it's 5x minus 4y is equal to 1.
- And let's say we have negative 10x plus 8y is equal to
- negative 30.
- Once again, I'm tempted to do elimination here, because I
- have a negative 10x and I have a 5x.
- If I take this top equation and I multiply it by 2, I'll
- get 10x minus 8y is equal to 2, right?
- I just multiplied both sides by 2.
- And if we add the left-hand sides, we'll get 0x plus 0y is
- equal to negative 28.
- So we essentially get 0 is equal to negative 28.
- That's crazy.
- We know that that's not true.
- This can never be true.
- We're getting an inconsistent statement.
- We're getting a weirdo statement, and that's because
- this has no solution.
- When you solve a system of equation, it doesn't matter
- how you do it, whether it's through substitution or
- whether it's through elimination, like I did here.
- When you get one of these statements where 0 equals
- negative 28 or 5 is equal to 7 or two things that clearly
- don't equal each other, when they essentially have to equal
- each other in order for the system to work, we call that
- an inconsistent system.
- And it will have no solution.
- So what does it mean for both of these
- equations to have no solution?
- Let's actually graph these.
- I think you'll have a better feel for what it means not to
- have a solution.
- So the first equation is 5x minus 4y is equal to 1.
- Let me put it into slope-intercept form.
- So if we subtract 5x from both sides, we get negative 4y is
- equal to negative 5x plus 1.
- Now if we divide both sides by negative 4, you get y is equal
- to negative 5/4x minus 1/4, right?
- 1 divided by negative 4.
- So this is the first equation, right over here, in
- slope-intercept form.
- Now let me write the second equation in
- slope-intercept form.
- We have negative 10x plus 8y is equal to negative 30.
- Let's add 10x to both sides.
- You get 8y is equal to 10x minus 30, and let's divide
- both sides by 8.
- You'll get y is equal to 10/8, is the same thing as 5/4.
- 5/4x minus 30/8.
- 30/8 is the same thing as 15/4.
- Oh, and actually I made a mistake here.
- When we divide both sides of this equation by negative 4,
- negative 5 divided by negative 4 is positive 5/4.
- So I shouldn't have had a negative there.
- I almost made a blunder.
- So there should be no negative there.
- Then a 1 divided by negative 4, is negative 1/4.
- So going back to the two equations, what do you notice?
- Well, when you put them in slope-intercept form, they
- both have the exact same slope, 5/4.
- but they have different y-intercepts.
- So what would their graphs look like?
- Let me graph them.
- Let's say that that is my y-axis.
- That is my x-axis.
- This equation, its y-intercept is at 0 negative 1, 4.
- Maybe that's that point right there.
- And it goes up at 5/4x.
- That's a little bit more than one.
- It's a 1.25x.
- Every time you go to the right 1, you're going up 1.25.
- So this line is going to look something like this.
- I'm just drawing it rough.
- I want you to get the general idea.
- That's what that line looks like.
- Now this line, its y-intercept is at negative 15/4.
- 15/4 is what?
- 3 3/4.
- So if this is negative 1/4, its y-intercept is going to be
- way down here someplace.
- I'll do it in the same color.
- Way down here someplace.
- Let me continue my x-axis down.
- This would be at negative 15/4.
- But its slope is the exact same thing.
- Every time you go to the right, 1, you're going to go
- up by 5/4, so its slope is going to be
- the exact same thing.
- So what do you notice about these two lines?
- They are parallel.
- They have the same slope, different y-intercepts, so
- they will never intersect.
- These two lines will never intersect.
- Which means that there is no point on the coordinate plane
- on the x-, y-coordinate plane that satisfies
- both of these equations.
- Remember, this line represents all of the points that
- satisfied this equation.
- This line represents all of the points that
- satisfied that equation.
- Notice, no point satisfies both.
- There's no point of intersection and that's why
- this was an inconsistent system.
- Let's do one more.
- Let's say I have 4x plus 5y is equal to 0, and I have 3x is
- equal to 6y plus 4.5.
- Actually, let me do a slightly different one, because I want
- to show you all of the different types that we can
- see in systems of equations.
- Let me clear this.
- Let's say my first system is 3x minus 7y is equal to 1.
- And let's say my other equation in my system is
- negative 6x plus 14y is equal to negative 2.
- So, let's try to find the x's and y's that satisfy this
- equation here.
- And just for a change of pace, let's do some substitution.
- Although this is very tempting to do elimination here, let's
- do substitution.
- You get 3x.
- Let's solve it for x.
- Actually, let's just do elimination, because this is
- just so glaringly prepared for elimination, let's
- just do it that way.
- So let's multiply this top equation by 2.
- And what do we get?
- We get 6x minus 14y is equal to 2 right?
- I just multiplied every term on both sides by 2.
- And now let's add the left sides together.
- You get 0 plus 0.
- And on the right hand side, you get is equal to 0.
- You get 0 is equal to 0, which is always going to be true.
- This type of system is called a dependent system.
- So remember, when you get a nice clean solution, that's a
- consistent system.
- When you get something crazy like 0 equals 1, that's an
- inconsistent system.
- That means the lines are parallel.
- When you get 0 equals 0, or 1 is equal to 1 or anything like
- that, you're dealing with the dependent system.
- Which really means that these are the exact same lines, even
- though they might look a little bit different.
- And to verify that, let's put them both into
- slope-intercept form.
- So this top line, you have 3x minus 7y is equal to 1.
- Let's subtract 3x from both sides.
- You get negative 7y is equal to negative 3x plus 1.
- Now let's divide both sides by negative 7.
- You get y is equal to positive 3/7x minus 1/7.
- That's that first equation.
- Now let's put the second equation into
- slope-intercept form.
- You have negative 6x plus 14y is equal to negative 2.
- Let's add 6x to both sides of the equation.
- So you get 14y is equal to 6x minus 2.
- Then divide both sides of the equation by 14.
- You get y is equal to 6/14x minus 2/14.
- Well, this is the same thing.
- 6/14 is the same thing is 3/7x minus 1/7.
- Notice, they are really the exact same equation.
- So if you want to find x's and y's that satisfy both, let's
- think about it.
- Let's graph it.
- So if that is my coordinate plane, that's the y-axis, that
- is the x-axis.
- This graph is going to look something like this.
- I'm going to draw it very roughly.
- It might look something like that, where its slope is 3/7.
- So this line is going to look something like that.
- That line looks exactly like that.
- It's the same exact line.
- So when you say, well, what are the x's and y's that
- satisfy both of these equations?
- Well, it's every x and y that's on these points.
- It's that x and y.
- That coordinate.
- That coordinate.
- That coordinate.
- There are an infinite number of solutions.
- And when we use the word dependent -- because you can
- get to one of these equations from the other.
- These equations are dependent on each other.
- You can just scale one or the other and rearrange it, and
- they equal each other.
- So here you have infinite solutions.
- Anything that satisfies one line will satisfy the other.
- So you just pick an x.
- When x is 1, you get 3/7 minus 1/7 that's 2/7.
- So 1, 2/7 satisfies both equations.
- If you pick x is equal to 0.
- 0, negative 1/7 satisfies both equations.
- And you could pick an infinite number of values for x, solve
- for y, and those coordinates will satisfy both equations.
- Let me review this a little bit.
- So we started off just with the plain vanilla.
- When you actually get a solution that is consistent.
- These lines actually intersect in one point.
- Then you have the situation where you get something crazy,
- when you solve your system of equations.
- 0 is equal to negative 28.
- Definitely not true.
- This is an inconsistent system.
- It has no solution, which means that
- these lines are parallel.
- They never intersect.
- And then finally, if you get something that's always true--
- that's just kind of silly how true it is-- this is a
- dependent system.
- These are going to be the same line.
- Then you can verify it by putting both of them into
- slope-intercept form.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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