Non-linear systems of equations
Non-Linear Systems of Equations 1 Non-Linear Systems of Equations 1
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- Solve the system of equations by graphing.
- Check your solution algebraically.
- Let's graph each of these, and let's start-- let me find a
- nice dark color to graph these with.
- Let me graph this top equation in blue, this parabola.
- The first thing to think about is this going to be an upward
- opening-- one, how did I know it's a parabola?
- That's because it's a quadratic function: we have an
- x squared term, a second degree term, here.
- Then we have to think about: it is going to be upward
- opening, or downward opening parabola?
- You see that it's a negative coefficient in front of the x
- squared, so it's going to be a downward opening parabola.
- What is going to be its maximum point?
- Let's think about that for a second.
- This whole term right here is always going to be negative,
- or it's always going to be non-positive.
- x squared will be non-negative when you multiply it by a
- negative, so it's going to be non-positive.
- So, the highest value that this thing can take on is when
- x is going to be equal to 0-- the vertex of this parabola is
- when x is equal to 0, and y is equal to 6.
- So, x is equal to 0, and y is 1, 2, 3, 4, 5, 6.
- So that right there is the highest point of our parabola.
- Then, if we want, we can a graph a couple of other
- points, just to see what happens.
- So let's see what happens when x is equal to-- let me just
- draw a little table here-- 2, what is y?
- It's negative x squared plus 6.
- So when x is 2, what is y?
- You have 2 squared, which is 4, but you have negative 2
- squared, so it's negative 4 plus 6-- it is equal to 2.
- It's the same thing when x is negative 2.
- You put negative 2 there, you square it, then you have
- positive 4, but you have a negative there, so it's
- negative 4 plus 6 is 2.
- You have both of those points there, so 2 comma 2, and then
- you have a negative 2 comma 2.
- If I were to graph it, Let's try it with 3, as well-- if we
- put a 3 there, 3 squared is 9.
- It then becomes a negative 9 plus 3, it becomes negative 3,
- and negative 3 will also become a negative 3.
- Negative 3 squared is positive 9, you have a negative out
- front, it becomes negative 9 plus 6, which is negative 3.
- You have negative 3, negative 3, and then you have 3,
- negative 3.
- So those are all good points.
- Now we can graph our parabola.
- Our parabola will look something-- I was doing well
- until that second part --like that, and let me just do the
- second part.
- That second part is hard to draw-- let me do it from here.
- It looks something like that.
- We connect to this dot right here, and then
- let me connect this.
- So that it looks something like that.
- That's what our parabola looks like, and obviously it keeps
- going down in that direction.
- So that's that first graph.
- Let's graph this second one over here: y is equal to
- negative 2x minus 2.
- This is just going to be a line.
- It's a linear equation, and the highest degree here is 1.
- Our y-intercept is negative 2, so 0, 1, 2.
- Our y-intercept is negative 2.
- Our slope is negative 2.
- If we move 1 in the x direction, we're going to go 2
- in the y-direction, and if we move 2 in the x direction,
- we're going to move down 4 in the y direction.
- If we move back 2, we're going to move up 2 in the y
- direction, and it looks like we found one of our points of
- intersection.
- Let's just draw that line, so that line will look something
- like-- It's hard for my hand to draw that, but let me try
- as best as I can.
- This is the hardest part.
- It will look something like that right there.
- The question is, where do they intersect?
- One point of intersection does immediately pop out at us,
- because they asked us to do it graphically.
- That point right there, which is the point negative 2, 2.
- It seems to pop out at us, so this is the
- point negative 2, 2.
- Let's see if that makes sense.
- When you have the point negative 2, when you put x is
- equal to negative 2 here, negative 2 times negative 2 is
- 4 minus 2, and y is equal to 2.
- When you put negative 2 here, y is also equal to 2, so that
- makes sense.
- There's going to be some other point way out here where they
- also intersect.
- There's also going to be some other point way out here if we
- keep making this parabola.
- When y is equal to positive 4, and you have negative 16 plus
- 6, you get negative 10.
- So, positive 1, 2, 3, 4, and then you go down 10.
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
- That looks like that might be our other point of
- intersection, so let me connect this right there.
- Our other point of intersection
- looks to be right there.
- If we just follow this red line it looks like we
- intersect there.
- Let's verify that it works out.
- So 4, negative 10.
- We know that that's on this blue line, so let's see if
- it's on this other line.
- So negative 2 times 4 minus 2, that is negative 8 minus 2,
- which is equal to negative 10.
- The point 4, negative 10, is on both of them.
- When x is equal to 4, y is negative 10 for both equations
- here, so they both definitely work out.
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