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### Rectangular Coordinate System

- The Cartesian coordinate system is made of two number lines attached at zero. One runs horizontally (
*x*-axis) and the other vertically (*y*-axis). - A point in this system is an ordered pair. The first value is an
*x*value and the second value a*y*value. They can be plotted by putting a point at the intersection of the*x*and*y*value from the respective axis. - Equations can be graphed by plotting the relation of x and y values. This is done by choosing values for one variable, and solving for the other variable.
- The point where the graph crosses the x-axis is known as the x-intercept. The point where the graph crosses the y-axis is known as the y-intercept.
- To determine if a point is on the graph of an equation it can be substituted into the equation. If it creates a true statement, it is on the graph.

### Rectangular Coordinate System

− 5x + 4y = 22

14 = 2xy + 5y + 9

6s = 10 − [2/t]

- t(6s) = ( 10 − [2/t] )t
- 6st = 10t − 2
- = 10t − 2

[4/x] + [5/y] = 0

[x/2] − [y/3] = 5

- For the x intercept, let y = 0[x/2] − [0/3] = 5
- [x/2] − 0 = 5
- [x/2] = 5
- x = 10

x intercept = 10 - For the y intercept, let x = 0

[0/2] − [y/3] = 5 - 0 − [y/3] = 5
- − [y/3] = 5
- − y = 15
- y = 15

y intercept = 15

y intercept = 15

[g/7] − [h/21] = 5

- For the g intercept, let h = 0

[g/7] − [0/21] = 5 - [g/7] − 0 = 5
- [g/7] = 5
- g = 35

g intercept = 35 - For the h intercept, let g = 0

[0/7] − [h/21] = 5 - 0 − [h/21] = 5
- − [h/21] = 5
- − h = 105
- h = − 105

h intercept = 105

h intercept = 105

[2x/9] − [y/4] = 12

- For the x intercept, let y = 0

[2x/9] − [0/4] = 12 - [2x/9] − 0 = 12
- [2x/9] = 12
- 2x = 108
- x = 54
- For the y intercept, let x = 0

[2(0)/9] − [y/4] = 7 - [0/9] − [y/4] = 7
- 0 − [y/4] = 7
- − [y/4] = 7
- − y = 28
- y = − 28

y intercept = −28

2x − 5y = 20

- For the y intercept, let x = 0
- 2(0) − 5y = 20
- − 5y = 20
- y = − 4
- For the x intercept, let y = 0
- 2x − 5(0) = 20
- 2x = 20
- x = 10

y intercept = −4

- Note the tick mark intervals

- Note the tick mark intervals

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Rectangular Coordinate System

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:11
- The Rectangular Coordinate System 0:39
- The Cartesian Coordinate System
- X-Axis
- Y-Axis
- Origin
- Quadrants
- Ordered Pairs
- Example 1 2:55
- The Rectangular Coordinate System Cont. 6:09
- X-Intercept
- Y-Intercept
- Relation of X-Values and Y-Values
- Example 2 11:03
- Example 3 12:13
- Example 4 14:10
- Example 5 18:38

### Algebra 1 Online Course

### Transcription: Rectangular Coordinate System

*Welcome back to www.educator.com.*0000

*In this lesson we are going to take a look at the rectangular coordinates system.*0002

*It will help us out as we get into more of the graphing process with some of equations later.*0006

*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.*0012

*We will learn all about the x axis, y axis, origin, quadrants and how we can plot an ordered pair.*0018

*When we get into plotting more things such as the graph we will look at looking at a table to do that graph.*0026

*And again how you can find the x and y intercepts of that line once on the graph.*0032

*The Cartesian coordinate system is formed by taking 2 number lines and putting them together after 0.*0041

*You will notice that one is in the up and down direction and one is completely horizontal.*0047

*Now the horizontal one, the one that goes completely flat, this guy right here, we will call this one our x axis.*0053

*The one that go straight up and down that will be known as our y axis.*0064

*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.*0071

*The little tick marks that you see usually represent how far you have to go on each of these number lines.*0087

*You can see that it breaks down into these number lines into so much larger parts of this graph.*0095

*We will give each of these squares a name, we will call them quadrants.*0103

*We usually number these quadrants starting in the upper right corner, say quadrant 1 and move in a counterclockwise direction.*0110

*This entire space here would be quadrant 2 but I have quadrant 3 and quadrant 4.*0118

*Now in each of these quadrants we can plot a point.*0126

* A point is an ordered pair and we look at it as a pair of the x and y put together.*0131

*If I'm looking at a point say out here, I have identify what it is x and y value are*0138

*by looking at how may tick marks I have to go over on the x axis and this would be 2.*0148

*How may tick marks I have to go in the y direction, 1, 2, 3, 4, I can plot out that point.*0154

*Keep in mind that with all of these points that you see, the first one is the x and the second one is y.*0163

*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.*0177

*In all of these, the first one will represent an x value and the second will represent our y value.*0185

*We will start at the origin with each of these and moving the x and then the y direction.*0191

*Starting with first one, we need to plot 2, 4.*0198

*I’m going to start at the origin and I'm going to move 2 units to the right and 4 units up.*0201

*This point right here will be the point 2, 4.*0213

*The next point has a negative value in it and it is a negative in the x.*0222

*I actually move left 1, 2, 3 and then I will go up 7, that point is located right here, 4, -3, 7.*0227

*As long as you can keep track of which one was x and which ones what is y, not a bad task at all.*0244

*The next one is -1 and -5, at we will move in the x direction -1 and then down 5.*0251

*1, 2, 3, 4, 5, so there is our point -1, -5.*0258

*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.*0269

*And one last one to go, this one is 0, 4.*0286

*What should you do if 0, should you go right or should you go left?*0293

*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.*0297

*Up 1, 2, 3, 4 that point is right here 0, 4.*0304

*Now that we have identified some points on here, we can actually say which quadrants each of these are in.*0313

*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.*0319

*The next point -3, 7 is in quadrant number 2.*0331

*-1, -5 quadrant number 3.*0338

*3, -2 that one is in quadrant number 4.*0345

*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.*0351

*Some more things about this rectangular coordinate system, when we get into actually trying to graph an equation on here*0370

*we can look at it as a special relation between its x values and its y values.*0377

*There are a lot of good things we can say about that equation.*0383

*To actually visually see what that relationship is and of course put it on the graph,*0386

*we will go ahead and create a table of values and see how they are related.*0390

*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.*0395

*Here are very important terms you want to keep in mind.*0402

*Where the graph of that equation crosses the x axis that will be known as its x intercept.*0404

*In a similar fashion, if it crosses the y axis, we will call that the y intercept.*0415

*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.*0422

*It still crosses the y axis so it has a y intercept.*0440

*You want to be able to identify both of these points.*0446

*A little bit more about generating those points from the table.*0453

*I said that a line is a special relationship between its x values and its y values and to see that relationship,*0457

*we want to be able to create a little chart and put in some values for either x or y.*0463

*Let us look at one I have 5x - 3y =12.*0476

*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.*0486

*What point should be there?*0493

*A good way to figure out if the point is on the graph or not, is to see if it simply satisfies the equation.*0495

*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.*0502

*And see if 3, 1 is on the graph or not.*0513

*(5 × 3) – (3 × 1) does that equal 12, let us find out.*0516

*That will be 15 - 3 = 12 or 12 = 12.*0524

*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.*0532

*Let us pick another point, 0, -4.*0546

*But when I plugged this one in, I get 5 × 0 which is 0 - 3 × - 4 = 12.*0554

*I know that one is on the graph as well, 0, -1, 2, 3, 4.*0570

*In practice, we usually do not pick up points out of thin air and then test to see if they work or not.*0583

*Usually we end up picking a lot of different things for one variable then see what it has to be for the other variable.*0588

*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.*0595

*Let us pick out some more values for x.*0603

*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.*0606

*That will be 10 - 3y = 12, I will subtract the 10 from both sides and get y = -2/3.*0620

*I know sure enough that is another point that I can put on my graph, 2 in the x direction then down 2/3.*0632

*What you may notice is I start to do more and more points over here on the graph.*0641

*You will get a better sense of what the entire graph looks like.*0645

*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.*0650

*Let us get into some more examples and see what other parts of the graph we can identify*0665

*and get a little bit more into the nuts and bolts of using that chart to graph out an entire line.*0669

*In this example we just simply want to find the x intercepts and the y intercept of the line.*0676

*Remember, this is where it crosses the x axis and where crosses the y axis.*0681

*My x axis is horizontal and I can see it crosses right at that point.*0686

*That would be at -1, 2, 3, 4, 5 in the x direction and 0 in the y direction.*0691

*x intercepts is at -5, 0 it crosses the y axis as well right up here at 4.*0701

*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.*0717

*With these ones we want to determine whether the point is on the line or not.*0735

*We can figure this out by substituting the values in for x and y.*0739

*The first one will always be an x and the next one a y.*0746

*-2 + 3y = 21, let us put in the -3 for x and 5 for y.*0753

*-2 × -3 = 6, 3 × 5 =15, for a total of 21, 6 + 15 = 21.*0768

*It looks like this one checks out since it makes the equation true, we can say that -3, 5 is on the line.*0783

*Let us do the same thing for the next one, my x is 1 and my y is 7.*0794

*Let see if this one rings true.*0811

*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.*0815

*What is that tells us that 1, 7 is not on the line.*0836

*You can take a point, plug it into the equation and see if it satisfies it and makes it true.*0845

*Let us get into more of the graphing process.*0852

*Here is an entire equation and I want to know what its entire graph looks like.*0854

*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.*0858

*We will generate them using a nice little table.*0867

*I will pick some values for x and we will see what makes y.*0870

*To make this processing a little bit easier, we will take our equation and we will solve for y first.*0875

*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.*0881

*This is the same equation, I just manipulated it a little bit so I can work with it a little bit easier.*0905

*We are going to pick some values for x, plug them in and see what they make y.*0912

*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.*0918

*Let us choose -3, -2, -1, 0, 1, 2 and 3.*0927

*With each of these, imagine what happens if you take -3 plug it in into the equation, what result do you get.*0938

*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.*0944

*-2/3 × -3 that would give us 2, 2 + 4 =6.*0955

*You put that on the other side, I know that -3, 6 is one of the points on our graph.*0966

* -3, 1, 2, 3, 4, 5, 6 there is a point right up there.*0972

*Let us take up the -3 and try this again, this time we will plug in -2.*0982

*This would give us 4/3 + 4 which should be the same as 5 and 1/3.*0991

*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.*1001

*-2 up 1, 2, 3, 4, 5 and just a third more there is that point right there.*1009

*Let us take out the -2 and put in -1 and let us see if this works out.*1019

*negative × negative would be a positive, 2/3 + 4 = 4 2/3.*1032

*-1 up 1, 2, 3, 4 and 2/3 right above there.*1043

*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.*1049

*We will do one more then will go ahead and connect the dots.*1058

*Let us see what happens if x =0, 0 × -2/3 would be 0, 0 + 4 = 4.*1066

*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.*1078

*Using this table we can definitely see a lot of points even sometimes we do not need to graph all the points,*1094

*we should graph enough of them that you have a good sense of what it looks like.*1100

*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.*1104

*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.*1110

*In this next one, we want to graph the equation again using a table of values.*1121

*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.*1126

*I will move that 5x together just by adding 5x to both sides then I will go ahead and divide by 2.*1133

*This is the equation that I will be using it is the same as the original.*1149

*I have just manipulated it a little bit so I will have an easier time working with it.*1153

*Let us pick some values to put in there for x so we can see what y needs to be.*1158

*We will pick some nice ones like -4, -2, 0, 2 and 4.*1165

*This one I’m picking values that are multiples of 2 since all of them we have to multiply by 5/2 first.*1172

*That will make our lives a little bit easier.*1179

*y =5/2, let us plug back in -4 then we will add 10.*1184

*-4 × 5/2 =-10 when we add 10 to that we get 0 so our first point is -4, 0.*1194

*Let us try another one, this one is -2, I will plug that one in there -2 × 5/2 =-5.*1212

*I will add 10 to that and get a total of 5 so this point -2, 5.*1229

*What happens if we go ahead and put in a 0, 0 × 5/2 = 0 + 10 = 10.*1250

*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.*1263

*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.*1275

*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.*1286

*If I put in 2 I will have 5 +10 = 15.*1293

*If we put in 4, 4 × 5/2 = 10 + 10 =20.*1305

*Now we have plenty of points, let us go ahead and graph this out.*1325

*I will go ahead and connect the dots.*1328

*The graph of this equation looks like this and with this one we have taken it definitely farther and mark out its x intercept.*1332

*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.*1342

*We will look at more next time with being able to do some shortcuts for graphing these lines*1362

*But with many types of equations you can often use a table to generate some values and put on some points.*1367

*Thank you for watching www.educator.com*1373

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 7:10 PM

Post by Ricardo Marquez on February 5, 2016

This is a great site but i would like to know if there is a topic of percent word problems.

Thanks

2 answers

Last reply by: Rafael Wang

Fri Sep 2, 2016 8:58 PM

Post by Ardeshir Badr on January 3, 2015

I have noticed a few mistakes in your Practice Questions-Answers and have compiled the screenshots concerned. Where can I send them?

Thank you