WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at the rectangular coordinates system.
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It will help us out as we get into more of the graphing process with some of equations later.
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Some of the specific things that we will look at as we are looking at our Cartesian coordinate system are just some of the parts of it.
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We will learn all about the x axis, y axis, origin, quadrants and how we can plot an ordered pair.
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When we get into plotting more things such as the graph we will look at looking at a table to do that graph.
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And again how you can find the x and y intercepts of that line once on the graph.
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The Cartesian coordinate system is formed by taking 2 number lines and putting them together after 0.
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You will notice that one is in the up and down direction and one is completely horizontal.
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Now the horizontal one, the one that goes completely flat, this guy right here, we will call this one our x axis.
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The one that go straight up and down that will be known as our y axis.
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The 0s of each of them when it both connects, we will also give that one a special name, we will call this point right here the origin.
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The little tick marks that you see usually represent how far you have to go on each of these number lines.
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You can see that it breaks down into these number lines into so much larger parts of this graph.
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We will give each of these squares a name, we will call them quadrants.
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We usually number these quadrants starting in the upper right corner, say quadrant 1 and move in a counterclockwise direction.
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This entire space here would be quadrant 2 but I have quadrant 3 and quadrant 4.
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Now in each of these quadrants we can plot a point.
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A point is an ordered pair and we look at it as a pair of the x and y put together.
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If I'm looking at a point say out here, I have identify what it is x and y value are
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by looking at how may tick marks I have to go over on the x axis and this would be 2.
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How may tick marks I have to go in the y direction, 1, 2, 3, 4, I can plot out that point.
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Keep in mind that with all of these points that you see, the first one is the x and the second one is y.
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Let us just do a real quick example of how we know little bit more about the parts of the graph and see if we can plot out some of these ordered pairs.
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In all of these, the first one will represent an x value and the second will represent our y value.
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We will start at the origin with each of these and moving the x and then the y direction.
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Starting with first one, we need to plot 2, 4.
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I’m going to start at the origin and I'm going to move 2 units to the right and 4 units up.
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This point right here will be the point 2, 4.
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The next point has a negative value in it and it is a negative in the x.
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I actually move left 1, 2, 3 and then I will go up 7, that point is located right here, 4, -3, 7.
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As long as you can keep track of which one was x and which ones what is y, not a bad task at all.
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The next one is -1 and -5, at we will move in the x direction -1 and then down 5.
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1, 2, 3, 4, 5, so there is our point -1, -5.
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Just a couple of more to do, the next one is 3, -2, that is in the positive direction 1, 2, 3 and then down 1, 2.
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And one last one to go, this one is 0, 4.
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What should you do if 0, should you go right or should you go left?
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Since it is at 0, we do not go right or left in the x direction, we actually stay at the origin and then we will move in the y direction 4.
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Up 1, 2, 3, 4 that point is right here 0, 4.
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Now that we have identified some points on here, we can actually say which quadrants each of these are in.
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If we look at our first point up here, our 2, 4 it is in quadrant number 1 because it is in that first square.
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The next point -3, 7 is in quadrant number 2.
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-1, -5 quadrant number 3.
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3, -2 that one is in quadrant number 4.
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The last point 0, 4 as the tricky one is not in quadrant 1, it is not in quadrant 2, it is actually right on the y axis.
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Some more things about this rectangular coordinate system, when we get into actually trying to graph an equation on here
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we can look at it as a special relation between its x values and its y values.
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There are a lot of good things we can say about that equation.
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To actually visually see what that relationship is and of course put it on the graph,
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we will go ahead and create a table of values and see how they are related.
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Usually this is done by picking a lot of different values for one of the variables and see what the other corresponding other values need to be.
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Here are very important terms you want to keep in mind.
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Where the graph of that equation crosses the x axis that will be known as its x intercept.
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In a similar fashion, if it crosses the y axis, we will call that the y intercept.
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Just draw a general graph so this is not a line since it is all curvy but it still crosses the x axis in some spot so I will call this the x intercept over there.
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It still crosses the y axis so it has a y intercept.
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You want to be able to identify both of these points.
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A little bit more about generating those points from the table.
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I said that a line is a special relationship between its x values and its y values and to see that relationship,
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we want to be able to create a little chart and put in some values for either x or y.
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Let us look at one I have 5x - 3y =12.
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If I want to go ahead and try and visualize what this equation looks like, I need to know what values go on my graph.
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What point should be there?
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A good way to figure out if the point is on the graph or not, is to see if it simply satisfies the equation.
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If I’m just picking some point like 3, 1, I can substitute the 3 in for x and 1 for y and see if it does satisfy it.
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And see if 3, 1 is on the graph or not.
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(5 × 3) – (3 × 1) does that equal 12, let us find out.
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That will be 15 - 3 = 12 or 12 = 12.
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What that tells me is that 3, 1 is one of the points on my graph, 1, 2, 3 on the x axis up one.
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Let us pick another point, 0, -4.
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But when I plugged this one in, I get 5 × 0 which is 0 - 3 × - 4 = 12.
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I know that one is on the graph as well, 0, -1, 2, 3, 4.
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In practice, we usually do not pick up points out of thin air and then test to see if they work or not.
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Usually we end up picking a lot of different things for one variable then see what it has to be for the other variable.
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Let us get a little bit of space in here and see how we would do this more in practice rather than just picking things out of thin air.
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Let us pick out some more values for x.
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For example what if x was 2, what with that 4y to be, we can find out by putting in the 2 for x and actually solving for y.
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That will be 10 - 3y = 12, I will subtract the 10 from both sides and get y = -2/3.
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I know sure enough that is another point that I can put on my graph, 2 in the x direction then down 2/3.
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What you may notice is I start to do more and more points over here on the graph.
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You will get a better sense of what the entire graph looks like.
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I have done about 3 points here but I’m going to draw lines for all of them to say that if I had even more points they would all line up along there.
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Let us get into some more examples and see what other parts of the graph we can identify
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and get a little bit more into the nuts and bolts of using that chart to graph out an entire line.
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In this example we just simply want to find the x intercepts and the y intercept of the line.
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Remember, this is where it crosses the x axis and where crosses the y axis.
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My x axis is horizontal and I can see it crosses right at that point.
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That would be at -1, 2, 3, 4, 5 in the x direction and 0 in the y direction.
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x intercepts is at -5, 0 it crosses the y axis as well right up here at 4.
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For that one we would not go in the x direction whatsoever but we would go out 4, so its y intercept is at 0, 4.
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With these ones we want to determine whether the point is on the line or not.
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We can figure this out by substituting the values in for x and y.
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The first one will always be an x and the next one a y.
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-2 + 3y = 21, let us put in the -3 for x and 5 for y.
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-2 × -3 = 6, 3 × 5 =15, for a total of 21, 6 + 15 = 21.
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It looks like this one checks out since it makes the equation true, we can say that -3, 5 is on the line.
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Let us do the same thing for the next one, my x is 1 and my y is 7.
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Let see if this one rings true.
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Plugging in 1 for x, 7 for y and simplify 3 + 4 is 7 and 5 × 7 is 35, these things are definitely not equal.
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What is that tells us that 1, 7 is not on the line.
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You can take a point, plug it into the equation and see if it satisfies it and makes it true.
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Let us get into more of the graphing process.
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Here is an entire equation and I want to know what its entire graph looks like.
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Rather than just picking points out of thin air and testing them, we are going to try and create a few values of our own by generating them.
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We will generate them using a nice little table.
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I will pick some values for x and we will see what makes y.
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To make this processing a little bit easier, we will take our equation and we will solve for y first.
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Our equation is 2x + 3y =12, to solve this for y I would move the 2x to the other side by subtracting a 2x and then divide it by 3, I will get -2/3x + 4.
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This is the same equation, I just manipulated it a little bit so I can work with it a little bit easier.
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We are going to pick some values for x, plug them in and see what they make y.
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It does not matter what values you pick but I do recommend choosing some negative values or may be throw in 0 and also choose some positive values.
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Let us choose -3, -2, -1, 0, 1, 2 and 3.
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With each of these, imagine what happens if you take -3 plug it in into the equation, what result do you get.
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If you do some scratch work that is okay, y = -2/3 we multiply that by -3, add 4 to it and see what the result is.
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-2/3 × -3 that would give us 2, 2 + 4 =6.
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You put that on the other side, I know that -3, 6 is one of the points on our graph.
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-3, 1, 2, 3, 4, 5, 6 there is a point right up there.
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Let us take up the -3 and try this again, this time we will plug in -2.
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This would give us 4/3 + 4 which should be the same as 5 and 1/3.
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It is okay if you get fractions with these just the way it turns out since you are choosing all kinds of different values for x.
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-2 up 1, 2, 3, 4, 5 and just a third more there is that point right there.
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Let us take out the -2 and put in -1 and let us see if this works out.
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negative × negative would be a positive, 2/3 + 4 = 4 2/3.
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-1 up 1, 2, 3, 4 and 2/3 right above there.
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I have not even graphed all of the points yet and you can see that they all seem to be following along in a straight line.
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We will do one more then will go ahead and connect the dots.
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Let us see what happens if x =0, 0 × -2/3 would be 0, 0 + 4 = 4.
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We will put that point on there as well right up here at 4 and now we will just connect all of our dots and this would be our graph of the equation.
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Using this table we can definitely see a lot of points even sometimes we do not need to graph all the points,
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we should graph enough of them that you have a good sense of what it looks like.
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Also if your graphing a line and you are going through this table and you got one that is a little off from all the rest, it is okay.
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But if it is off from all the rest then we will check your work on now to make sure that you have done it correctly.
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In this next one, we want to graph the equation again using a table of values.
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Like last time, I will go ahead and solve this for y so we have a little bit easier of the time working with it.
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I will move that 5x together just by adding 5x to both sides then I will go ahead and divide by 2.
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This is the equation that I will be using it is the same as the original.
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I have just manipulated it a little bit so I will have an easier time working with it.
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Let us pick some values to put in there for x so we can see what y needs to be.
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We will pick some nice ones like -4, -2, 0, 2 and 4.
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This one I’m picking values that are multiples of 2 since all of them we have to multiply by 5/2 first.
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That will make our lives a little bit easier.
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y =5/2, let us plug back in -4 then we will add 10.
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-4 × 5/2 =-10 when we add 10 to that we get 0 so our first point is -4, 0.
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Let us try another one, this one is -2, I will plug that one in there -2 × 5/2 =-5.
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I will add 10 to that and get a total of 5 so this point -2, 5.
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What happens if we go ahead and put in a 0, 0 × 5/2 = 0 + 10 = 10.
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I have another point for my chart here and this one is off the chart 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 way up here.
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We can already see that after doing a bunch of these, the rest of my graph is going to go completely off of my graph paper.
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We will go ahead and do the other 2 just so you can see what their value will be but we already have plenty of points so we go ahead and graph these line for 1, 2.
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If I put in 2 I will have 5 +10 = 15.
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If we put in 4, 4 × 5/2 = 10 + 10 =20.
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Now we have plenty of points, let us go ahead and graph this out.
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I will go ahead and connect the dots.
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The graph of this equation looks like this and with this one we have taken it definitely farther and mark out its x intercept.
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This one is at -4, 0 and its y intercept way up here will be at 0, 10 and see they show up in the chart here and here when one of the variables is 0.
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We will look at more next time with being able to do some shortcuts for graphing these lines
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But with many types of equations you can often use a table to generate some values and put on some points.
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