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### Systems of Linear Equations

- A system of equations is made from a pair or more equations put together. If all the equations are lines, it is said to be a linear system.
- A solution to a system must make all equations true.
- To solve a system graphically, you plot out each equation and identify where they cross. For this method to work you must draw the graphs accurately.
- A system in two variables could have one solution, no solution, or infinite solutions. You can recognize these by the lines either crossing at one point, being parallel, or being on top of one another.

### Systems of Linear Equations

x + 2y = 3

5x + y = 5

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = 3 → ( 0,[1/2] ),( 3,0 )
- 5x + y = 5 → ( 5.0, ),( 5,0 )

(7, - 2)

y - x = 4

2y - x = 3

- Find two points for each equation by setting x = 0 then y = 0
- y − x = 4 → ( 0,4 ),( − 4,0 )
- 2y − x = 3 → ( 0,[3/2] ),( − [3/2],0 )

( - 5, - 1)

- x + 4y = 20

y - 3x = 16

- Find two points for each equation by setting x = 0 then y = 0
- x + 4y = 20 → ( 0,5 ),( − 20,0 )
- y − 3x = 16 → ( 0, − [16/3] ),( 16,0 )

( - 4,4)

- x + 6y = 12

y + x = 9

- Find two points for each equation by setting x = 0 then y = 0
- − x + 6y = 12 → ( 0, − 12 ),( 2,0 )
- y + x = 9 → ( 9,0 ),( 0,9 )

(6,3)

x + 2y = 16

x + y = 8

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = 16 → ( 0,8 ),( 16,0 )
- x − y = 8 → ( 8,0 ),( 0,8 )

(0,8)

x + 2y = - 12

3y - x = 6

- Find two points for each equation by setting x = 0 then y = 0
- x + 2y = − 12 → ( − 12,0 ),( 0, − 6 )
- 3y − x = 6 → ( − 6,0 ),( 0,2 )

( - 6,0)

x + y = 5

y = - 3

- Find two points for each equation by setting x = 0 then y = 0
- x + y = 5 → ( 5,0 ),( 0,5 )
- y = − 3 → ( 0, − 3 ),( 0, − 3 )

(8, - 3)

y - 3x = 2

- 9x + 3y = 6

- Find two points for each equation by setting x = 0 then y = 0
- x + y = 5 → ( − [3/2],0 ),( 0,2 )
- y = − 3 → ( − [3/2],0 ),( 0,2 )

Infinite solutions

2y - x = - 8

y - x = - 3

- Find two points for each equation by setting x = 0 then y = 0
- 2y − x = − 8 → ( 0, − 4 ),( 8,0 )
- − y + x = 3 → ( 3,0 ),( 0, − 3 )

( - 2, - 5)

y + 2x = 5

x + [y/2] = 2

- Find two points for each equation by setting x = 0 then y = 0
- y + 2x = 5 → ( 0,5 ),( [5/2],0 )
- x + [y/2] = 2 → ( 2,0 ),( 0,4 )

No solutions

*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

### Systems of Linear Equations

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 0:13
- Systems of Linear Equations 0:46
- System of Equations
- System of Linear Equations
- Solutions
- Points as Solutions
- Finding Solutions Graphically
- Example 1 6:37
- Example 2 12:07
- Systems of Linear Equations Cont. 17:01
- One Solution, No Solution, or Infinite Solutions
- Example 3 18:31
- Example 4 22:37

### Algebra 1 Online Course

### Transcription: Systems of Linear Equations

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

*In this lesson we are going to take a look at systems of linear equations, pairs of linear equations*0002

*and get into their solutions and how we can graph them.*0008

*More specific things that we want to know is, what exactly is a system of equation and how can we find their solutions?*0015

*One of the greatest ways that we can do to find their solutions is just looking at their graphs.*0023

*Some special things that we want to know in terms of their types of solutions is how do we know when there is going to be no solution*0030

*and how we will know when they will actually be an infinite amount of solutions.*0036

*Watch out for those two as we get into the nuts and bolts of all this.*0041

*What is a system of equations?*0049

*A system is a pair or more of an equation coupled together.*0052

*Sometimes you will see a { } put onto equations just to show that they are all being combined and coupled together.*0059

*Here I have a pair, but I could have 3, 4, 5 and many more than just that.*0070

*Now if all of the equations that are being coupled together are lines, then we will call these linear equations.*0076

*You will notice in the examples that I have up here, both of these are lines in standard form.*0083

*I have a system of linear equations.*0089

*In order to be a solution of a system, it must satisfy all of the equations in that system all simultaneously, all at once.*0094

*If it only satisfies the first equation then that is not good enough to be a solution of the system, it must satisfy all of them.*0101

*Let us quickly look at how we can determine if a point is a part of the solution or not.*0116

*We are going to do this by substituting it into an actual system.*0123

*My system over here is 2x + 3y = 17 and x + 4y = 21.*0126

*Is the point 7, 1 a solution or not?*0133

*Let us go ahead and write down our system.*0138

*Let us first put it into equation 1, we have (2 × 7) + (3 × 1) is it equal to 17, I do not know.*0144

*Let us find out.*0157

*2 × 7 would be 14 + 3 and sure enough I get 17.*0158

*I know it satisfies the first equation just fine.*0168

*Let us take it and substitute it into this second equation.*0174

*I have 7x + (4 × y value × 1).*0182

*4 × 1 = 4, 7 + 4, that one does not work out because 11 is not equal to 21.*0193

*Even though it only satisfies one of the equations, I can say that it is not a solution.*0206

*It only satisfies one equation.*0223

*Let us try this with the point 1, 5 and let us see if that is a solution of the system.*0234

*We will put in 2 or 3 if we want to know if this is equal to 17.*0241

*I will put in 1 for x and we will put in 5 for y and let us simplify it.*0248

* 2 × 1 is 2 ,3 × 5 is 15 and 2 + 15 =17 and this one checks out.*0258

*Now let us substitute it into the second equation.*0273

*I will take 1 and put in for x, we will take 5 and put it in for y.*0278

*I have 1 + 20 this that equal 21? It does 21 equals 21.*0287

*I can say it is a solution and it satisfies both of the equations.*0296

*There are few ways that you can go about solving a system of equations and we will see them in later lessons*0316

*or the first ways that you can sense what the solution should be is to the graph the equations that are present.*0321

*If these are linear equations that you are dealing with then you can use the techniques of graphing your lines to make this a much easier process.*0328

*Let us work on finding our solutions graphically.*0335

*To do this, all you have to do is graph the equations and then find out where those equations cross, when they do that is going to be our solution.*0339

*There is one downside to this so be very careful.*0349

*In order for this process to work out, you must make accurate graphs.*0353

*If your lines are a little wavy or you do not make them very accurate then you might think you know where it cross, but actually identify a different point.*0358

*Be careful and make accurate graphs.*0369

*Since finding solutions using a graph is as nice and visual, we definitely will start with it*0381

*but we will use some of those later techniques for solving because the accuracy does tend to be an issue with the graphs.*0388

*Let us get into the graphing and finding a solution that way, I will try.*0400

*Here I have a system of linear equations and it looks like both of them are almost written in standard form.*0405

*The second one is not quite as in that form because it has a negative sign out front.*0412

*I think we will be able to graph it just fine.*0417

*I'm going to graph it by using its intercepts.*0419

*We will make a little chart for our first equation and I'm going to identify where it crosses the y axis and the x axis by plugging in an appropriate value of 0.*0424

*If I use 0 in my equation for the first one, I just need to solve and figure out what y is.*0436

*Putting in a 0, we will limit that term and solving the rest out, divide it by 3 and I will get that y =2.*0445

*I know that is going to be one point on my graph, 0, 2.*0457

*We will go ahead and plug in 0 for y and we can see that it will eliminate our y terms and now we just have to solve 2x = 6.*0467

*If we divide both sides of that by 2, we will get x =3.*0484

*We have a second point that we can go ahead and put on our graph, 3, 0.*0492

*Now that we have two points, let us be accurate on graphing this out.*0501

*If you want to make it even more accurate, one thing you can do is actually use more points to help you graph this out.*0517

*We have one line on here, it looks pretty good.*0526

*Let use another chart here and see if we can graph out the second one.*0528

*Once we have both lines on here, we will go ahead and take a look at where they cross.*0540

*This will be for line number two, we will get x and y. What happens when we put in some 0?*0545

*-0 + y = 7 that is a nice one to solve.*0554

*-0 is the same as 0, the only thing left is y = 7.*0560

*Let us put on the point 0, 7 right up there.*0567

*If we put in 0 for y, -x + 0 = 7, I will get that –x =7 or x = -7.*0578

*There is another point I can put on there.*0591

*Now that I have two points here, let us go ahead and connect the dots.*0599

*We can see where our two lines cross.*0614

*This is from our first line and the blue line is from our second equation.*0618

*Right here, it looks like they definitely cross.*0626

*That is the point -3, 1, 2, 3, 4.*0630

*The reason why we went over a little bit of work on testing solutions is, if our graph was inaccurate and this was incorrect,*0638

*we can take it and put it into our system just to double check that it actually works out.*0645

*If you want you can take -3 and 4, plug it back in and let us see what happens.*0650

*Starting with the first equation 2, 3y = 6 and let us plug in our x and y.*0664

*Checking to see if this is the solution, to see if it satisfies our first equation.*0678

* -6 + 12 does that equal 6? -6 + 12 does equal 6.*0683

*It checks out for our first equation.*0691

*Let us put it into the second one, our x value is -3 our y value is 4.*0696

*Negative times negative would be 3.*0710

*I’m sure enough 3 + 4 does equal 7 so it satisfies the second equation as well.*0716

*I know that -3, 4 is definitely my solution.*0721

*Let us try and find a solution to another system.*0730

*Here I have -2x + y = -8 and y = -3x + 2.*0735

*The first one I can go and ahead and make a chart for and maybe figure out to where its intercepts are.*0744

*That seems like a good way to find out that one.*0752

*What happens when x = 2, what value do we get for y?*0755

*This term would drop away because of the 0 and I have only be left with y = -8.*0765

*I know that is one point I will need.*0772

*Let us go ahead and put a dot on there.*0775

*x =0, y = -1, 2, 3, 4, 5, 6, 7, 8 way down here.*0777

*Now we put in a 0 for y, -2x + 0 = -8, let us see how this one turns out.*0786

*My 0 term will drop away and we continue solving for x by dividing both sides by -2, -8 ÷ -2 is 4.*0799

*There is another point, 4, 0, 1, 2, 3, 4, 0.*0818

*Now that we have two points, let us go ahead and connect them together and we will see our entire line.*0824

*We got one of them down and now let us graph our second one.*0843

*Notice how the second equation in our system is written in slope intercept form*0847

*which means we will take a bit of a short cut by using just the y intercept and also using its slope.*0852

*I will make things much easier.*0862

*Our y intercept is 2, I know it goes through this point right here.*0864

*Our slope is -3 which I can view as -3/1.*0868

*Starting at our y intercept we will go down 3 and to the right 1.*0873

*1, 2, 3 into the right 1, there is a point.*0878

*I’m bringing out my ruler and we will connect these.*0885

*Now we can check if each of them cross.*0897

*Let us highlight which equations go with which line, there we go.*0902

*Also, it looks like they do cross and I would say it crossed down here at 2, -4.*0906

*Even though I made some pretty accurate graphs, it does look like it is a hair larger than 2, and sometimes this happens.*0916

*Even when we do make some pretty accurate graphs, sometimes where they cross it looks like it is just a hair off.*0923

*The frustrating part is the actual solution may be different from 2, -4.*0932

*About the only way I'm sure when using this graphing method is to check it by putting it in to the system.*0937

*Later on we will look at more accurate systems where we would not have to check quite as much.*0944

*Let us go ahead and put in our values for x and y to see if it satisfies both equations.*0950

*x is 2 and y is -4, -2 × 2 = -4 + -4 is a -8.*0958

*It looks like the first one checks out, everything is nice and balanced so it satisfies the first equation.*0978

*Let us go ahead and put it into the second one and see if that one also works.*0986

*X is 2 and y is -4, let us see if this balances out.*0994

*-4 is not equal to -6 + 2 I think it is, because when you add -6 and 2 you will get -4.*1003

*It satisfies both equations.*1014

*2, -4 is a solution.*1017

*When you are going through finding different solutions for a system, several different things could happen.*1023

*The first situation that could happen is a lot like the examples we just covered.*1030

*You are going to go through the graphing process and you are going to find one spot where the two actually cross.*1035

*You will notice this kind of case that you like to be in.*1040

*However, when graphing these lines, you might also find that the lines are completely parallel.*1043

*It does pose a little bit of a problem because it means that the lines would not cross whatsoever.*1049

*With parallel lines, we will not have a solution to our system.*1065

*Another thing that could possibly happen is you go to graph the lines and they turn out to be exactly the same line.*1071

*In that situation, we will not only get solutions but we will get lots of solutions because if they are the same line, they will cross an infinite amount of spots.*1078

*Basically everywhere on a line will be a solution.*1087

*Watch for these to show up when you are making these graphs, you will either find one spot where they cross then you have a solution.*1090

*You will see that the lines are parallel and they do not cross so you have no solution.*1098

*You will find that they are exactly the same line and then you will have an infinite number of points as your solution.*1103

*In this next example, I want to highlight that the two special cases that could happen.*1114

*One thing I want to point out is that when you are looking at the system,*1119

*it is sometimes not obvious whether it has no solution or an infinite amount of solutions.*1122

*You got to get down into the graphing process or the solving process before you realize that.*1127

*My first equation is x +2, y= 4 and the other one is 3x + 6, y =18.*1133

*I’m going to graph these out using my x and y intercepts.*1140

*What happens when x =0, what happens when y= 0?*1146

*For the first one, I put in 0 for x and we will go ahead and we start solving for y.*1149

*You can see my 0 term is going to drop away, which is good.*1157

*I will divide both sides by 2 and I will get that y = 2.*1161

*There is one point that I will mark on my graph 0, 2.*1169

*I will put in 0 for y and let us see what happens with that one.*1179

*The 0 term is going to drop out and the only thing I'm left with is x = 4.*1189

*I will put that point in there, 4, 1, 2, 3, 4, 0.*1199

*I will go ahead and graph it out.*1204

*Now that we have one line, let us go ahead and graph the other.*1219

*We will get its intercepts by putting in these 0 for x and 0 for y.*1228

*(3 × 0) + 6 time to solve for y.*1243

*Our 0 term will go away and I will just be left with 6y =18.*1251

*6 goes into 18 three times so I have 0, 3 as one of my end points.*1259

*Putting in 0 for y, let us see what happens there.*1273

*3x + 6 × 0 = 18.*1275

*It looks like that term will drop away and I will just have 3x =18.*1282

*3 goes into 18 six times, x = 6.*1290

*1, 2, 3, 4, 5, 6 I have that one.*1300

*Let us go ahead and graph it out now.*1306

*What we can see from this one is that it looks like the two lines are parallel and they are not crossing whatsoever.*1320

*If it is a little off and it looks like one of your lines could potentially cross but maybe off the graph,*1330

*one thing you can do is you can rewrite the system into slope intercept form so you can better check the slopes.*1335

*I'm pretty sure that these are parallel, I’m going to say that there is no solution to the system.*1344

*They do not cross whatsoever.*1353

*Let us try one more and see if this one has a solution.*1359

*This is y = 4x - 4 and 8x -2y =8.*1362

*The first one is written in slope intercept form so I will graph it by identifying the y intercept and its slope.*1370

*This starts at -4 and starting at the - 4 I will go up 4/1and just like that we can go ahead and graph it out.*1382

*There is our first line.*1406

*The second one is written more in standard form so I will go ahead and use its intercepts to track that one down.*1408

*What happens when x =0 and what happens when y = 0.*1415

*When x =0, that will drop away that term right there and I'm looking at -2y = 8.*1422

*We will be dividing both sides by -2 and I will end up with -4.*1434

*There is one point I can put on there, notice it is on the same spot.*1443

*Putting in 0 for y, I can see the term that will go away.*1451

*I am left with 8x = 8.*1468

*Dividing both sides by 8, I am simply be left with x = 1.*1471

*We will put that point on there and both points ended up on the other line.*1482

*When I go to draw this out, you will see that one line actually ends up right on top of the other one.*1487

*They are essentially the same line.*1492

*In this case, we end up with an infinite amount of solutions because they still cross, but they cross in lots of different spots.*1501

*I could say they cross everywhere, there are infinite number of solutions.*1508

*One of the best ways that you can figure out a solution to a system of equations*1530

*is simply graph both of the equations that are present in the system and see where they cross.*1533

*We will look at some more accurate methods in the next lesson.*1539

*Thank you for watching www.educator.com.*1542

1 answer

Last reply by: Professor Eric Smith

Thu Dec 31, 2015 5:32 PM

Post by Bongani Makhathini on December 30, 2015

I believe that example 3 does have a solution, its just that the lines will across outside the graphing paper displayed on the screen as the lines progress. I came to this conclusion because the points 4, 2 and 6 are even numbers. However, 3 is an odd number, tilting the line just slightly anti-clockwise. :-)

0 answers

Post by Farhat Muruwat on March 24, 2014

Step 3. 2y âˆ’ x = 3 â†’ ( 0,[3/2] ),( âˆ’ [3/2],0 )

How did you get 2y - x = 3?

0 answers

Post by Farhat Muruwat on March 24, 2014

Q. Solving the system by graphing

y âˆ’ x = 42y âˆ’ x = 3

Step 1. Find two points for each equation by setting x = 0 then y = 0

Step 2. y âˆ’ x = 4 â†’ ( 0,4 ),( âˆ’ 4,0 )

How did you get y - x = 4?