Solving systems with elimination (addition-elimination)
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Example 1: Solving systems by elimination
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Example 2: Solving systems by elimination
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Addition Elimination Method 1
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Addition Elimination Method 2
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Addition Elimination Method 3
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Addition Elimination Method 4
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Example 3: Solving systems by elimination
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Simple Elimination Practice
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Systems of equations with simple elimination
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Systems with Elimination Practice
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Systems of equations with elimination
Example 3: Solving systems by elimination Solving systems by elimination 3
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- We're told to solve and graph the solution for the system of
- equations right here.
- And the first thing that jumps out at me, is that we might be
- able to eliminate one of the variables.
- And if we just focus on the x, we have a 4x here and we have
- a 2x right here.
- If we were to just add them right now, we would get a 6x.
- So that wouldn't eliminate it.
- But if we can multiply this 2x by negative 2, it'll become a
- negative 4x, and then when you add it, they would cancel out.
- So let's multiply this equation, this second
- equation, by negative 2.
- So I'm going to multiply both sides of this equation by
- negative 2.
- And the whole motivation is so that this 2x becomes a
- negative 4x.
- And, of course, I can't just multiply only the 2x.
- Anything I do to the left-hand side of the equation I have to
- do to every term, and I have to do to both
- sides of the equation.
- So the second equation becomes negative 4x-- that's negative
- 2 times 2x-- plus-- we have negative 2 times negative y--
- which is plus 2y is equal to 2.5 times negative 2, is equal
- to negative 5.
- I just rewrote the second equation, multiplying both
- sides by negative 2.
- Now, this top equation-- I'll write it on the bottom now--
- we have 4x minus 2y is equal to positive 5.
- And now we can eliminate it.
- We can say, hey, look, the negative 4x and the positive
- 4x should cancel out, or they will cancel out.
- So let's add these two equations.
- Let's add the left side to the left side, the right side to
- the right side, and we can do that because these
- two things are equal.
- We're doing the same thing to both sides of the equation.
- So what do we get?
- If we take our negative 4x plus our 4x, well, those
- cancel out.
- So you're left with nothing.
- Maybe I could write a 0 there.
- 0x if you want.
- And then you have your plus 2y and your negative 2y.
- Those also cancel out.
- So you're also left with 0y.
- And then that equals negative 5 plus 5 is equal to 0.
- So this just simplifies to 0 equals 0, which is true, but
- it's kind of bizarre.
- We had all these x's and y's.
- Everything canceled out.
- So let's explore this a little bit more.
- Let's graph it and see what this 0 equals 0 is telling us
- when we try to solve this system of equations.
- So let me graph this top guy.
- I'll do it in blue.
- So right now it's in standard form.
- Let's put it in slope-intercept form.
- So we have 4x minus 2y is equal to 5.
- Let's subtract 4x from both sides.
- I want the x terms on the right-hand side.
- So then I'm left with negative 2y is equal to
- negative 4x plus 5.
- Now we can divide both sides by negative 2.
- And we are left with y is equal to positive 2x, right,
- that's positive 2x, minus 2.5.
- So let's graph that.
- The y-intercept is negative 2.5.
- So negative 2.5 right there, and then it has a slope of 2.
- So if we move up 1, if we move up in the x-direction, if we
- move to the right 1 in the positive x-direction, we
- will move up 2.
- So 1, 2.
- Right there.
- And if we were to do it again, we move up 1, 2.
- Just like that.
- So the line's going to look something like this.
- I'll try my best to draw a straight line.
- This is the hardest part about a lot of these
- problems. There you go.
- So that's the top equation.
- Now, let me draw the bottom equation.
- Let me draw and I'll do it in this green color.
- So this bottom equation was 2x minus y is equal to 2.5.
- And we can subtract 2x from both sides.
- The left-hand side becomes negative y is equal to 2x
- plus-- or is equal to negative 2x plus 2.5.
- Now let's multiply or divide both sides by negative 1.
- And you get y is equal to positive 2x minus 2.5.
- And let's try to graph this, and you already might notice
- something interesting about these two equations.
- You try to graph this, the y-intercept is at negative
- 2.5, right there.
- The slope is 2.
- So it's going to be this exact same line.
- And you saw that algebraically.
- I didn't have to graph it.
- These two lines have the exact same equation when you put
- them in slope-intercept form.
- That's the first equation.
- That's the second equation.
- So what this 0 equals 0 is telling us is actually that
- these are the same line.
- That these actually have an infinite number of solutions.
- Any point on this line, which is both of those lines, will
- satisfy both of these equations.
- You give me an arbitrary y, solve for x in the top
- equation, that x and y will also
- satisfy the bottom equation.
- So this actually has an infinite number of solutions.
- These are the same line.
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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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