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
Solving Linear Systems by Graphing Solving Linear Systems by Graphing
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- Let's say I have the equation y is equal to x plus 3.
- And I want to graph all of the sets, all of the coordinates x
- comma y that satisfy this equation right there.
- And we've done this many times before.
- So we draw our axis, our axes.
- That's my y-axis.
- This is my x-axis.
- And this is already in mx plus b form, or
- slope-intercept form.
- The y-intercept here is y is equal to 3, and the
- slope here is 1.
- So this line is going to look like this.
- We intersect at 0 comma 3-- 1, 2, 3.
- At 0 comma 3.
- And we have a slope of 1, so every 1 we go to the
- right, we go up 1.
- So the line will look something like that.
- It's a good enough approximation.
- So the line will look like this.
- And remember, when I'm drawing a line, every point on this
- line is a solution to this equation.
- Or it represents a pair of x and y that
- satisfy this equation.
- So maybe when you take x is equal to 5, you go to the
- line, and you're going to see, gee, when x is equal to 5 on
- that line, y is equal to 8 is a solution.
- And it's going to sit on the line.
- So this represents the solution set to this equation,
- all of the coordinates that satisfy y is
- equal to x plus 3.
- Now let's say we have another equation.
- Let's say we have an equation y is equal to
- negative x plus 3.
- And we want to graph all of the x and y pairs that satisfy
- this equation.
- Well, we can do the same thing.
- This has a y-intercept also at 3, right there.
- But its slope is negative 1.
- So it's going to look something like this.
- Every time you move to the right 1, you're going
- to move down 1.
- Or if you move to the right a bunch, you're going to move
- down that same bunch.
- So that's what this equation will look like.
- Every point on this line represents a x and y pair that
- will satisfy this equation.
- Now, what if I were to ask you, is there an x and y pair
- that satisfies both of these equations?
- Is there a point or coordinate that satisfies both equations?
- Well, think about it.
- Everything that satisfies this first equation is on this
- green line right here, and everything that satisfies this
- purple equation is on the purple line right there.
- So what satisfies both?
- Well, if there's a point that's on both lines, or
- essentially, a point of intersection of the lines.
- So in this situation, this point is on both lines.
- And that's actually the y-intercept.
- So the point 0, 3 is on both of these lines.
- So that coordinate pair, or that x, y pair, must satisfy
- both equations.
- And you can try it out.
- When x is 0 here, 0 plus 3 is equal to 3.
- When x is 0 here, 0 plus 3 is equal to 3.
- It satisfies both of these equations.
- So what we just did, in a graphical way, is solve a
- system of equations.
- Let me write that down.
- And all that means is we have several equations.
- Each of them constrain our x's and y's.
- So in this case, the first one is y is equal to x plus 3, and
- then the second one is y is equal to negative x plus 3.
- This constrained it to a line in the xy plane, this
- constrained our solution set to another
- line in the xy plane.
- And if we want to know the x's and y's that satisfy both of
- these, it's going to be the intersection of those lines.
- So one way to solve these systems of equations is to
- graph both lines, both equations, and then look at
- their intersection.
- And that will be the solution to both of these equations.
- In the next few videos, we're going to see other ways to
- solve it, that are maybe more
- mathematical and less graphical.
- But I really want you to understand the graphical
- nature of solving systems of equations.
- Let's do another one.
- Let's say we have y is equal to 3x minus 6.
- That's one of our equations.
- And let's say the other equation is y is equal to
- negative x plus 6.
- And just like the last video, let's graph both of these.
- I'll try to do it as precisely as I can.
- There you go.
- Let me draw some.
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- And then 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- I should have just copied and pasted some graph paper here,
- but I think this'll do the job.
- So let's graph this purple equation here.
- Y-intercept is negative 6, so we have-- let me do another
- [? slash-- ?]
- 1, 2, 3, 4, 5, 6.
- So that's y is equal to negative 6.
- And then the slope is 3.
- So every time you move 1, you go up 3.
- You moved to the right 1, your run is 1, your
- rise is 1, 2, 3.
- That's 3, right?
- 1, 2, 3.
- So the equation, the line will look like this.
- And it looks like I intersect at the point 2
- comma 0, which is right.
- 3 times 2 is 6, minus 6 is 0.
- So our line will look something
- like that right there.
- That's that line there.
- What about this line?
- Our y-intercept is plus 6.
- 1, 2, 3, 4, 5, 6.
- And our slope is negative 1.
- So every time we go 1 to the right, we go down 1.
- And so this will intersect at-- well, when y is equal to
- 0, x is equal to 6.
- 1, 2, 3, 4, 5, 6.
- So right over there.
- So this line will look like that.
- The graph, I want to get it as exact as possible.
- And so we're going to ask ourselves the same question.
- What is an x, y pair that satisfies
- both of these equations?
- Well, you look at it here, it's going to be this point.
- This point lies on both lines.
- And let's see if we can figure out what that point is.
- Just eyeballing the graph here, it looks like we're at
- 1, 2, 3 comma 1, 2, 3.
- It looks like this is the same point right there, that this
- is the point 3 comma 3.
- I'm doing it just on inspecting my hand-drawn
- graphs, so maybe it's not the exact--
- let's check this answer.
- Let's see if x is equal to 3, y equals 3 definitely
- satisfies both these equations.
- So if we check it into the first equation, you get 3 is
- equal to 3 times 3, minus 6.
- This is 9 minus 6, which is indeed 3.
- So 3 comma 3 satisfies the top equation.
- And let's see if it satisfies the bottom equation.
- You get 3 is equal to negative 3 plus 6, and negative 3 plus
- 6 is indeed 3.
- So even with our hand-drawn graph, we were able to inspect
- it and see that, yes, we were able to come up with the point
- 3 comma 3, and that does satisfy
- both of these equations.
- So we were able to solve this system of equations.
- When we say system of equations, we just mean many
- equations that have many unknowns.
- They don't have to be, but they tend to have more than
- one unknown.
- And you use each equation as a constraint on your variables,
- and you try to find the intersection of the equations
- to find a solution to all of them.
- In the next few videos, we'll see more algebraic ways of
- solving these than drawing their two graphs and trying to
- find their intersection points.
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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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