### Linear Equations in Two Variables

- To build the equation of a line you can use point-slope form ( y – y1 = m(x-x1) ). To use this you must know the slope of the line and one point on the line.
- Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of one another.
- You can test if two lines are perpendicular by multiplying their slopes together. If you get -1 then you know they are perpendicular.
- You may have to use other formulas such as the slope formula, before using the point-slope form.

### Linear Equations in Two Variables

Find the equation of this line in point slope form.

- y − y
_{1}= m(x − x_{1}) - y − ( − 3) = m[x − ( − 3)]
- y + 3 = m(x + 3)

Find the equation of this line in point slope form.

- y − y
_{1}= m(x − x_{1}) - y − ( − 1) = − 7(x − 11)

Find the equation of this line in point slope form.

- y − y
_{1}= m(x − x_{1}) - y − 10 = − 8[x − ( − 14)]

- slope = 0 for a horizontal line

m = 0 - y − y
_{1}= m(x − x_{1})

- slope = 0 for a horizontal line

m = 0 - y − y
_{1}= m(x − x_{1}) - y − ( − 13) = 0[x − ( − 12)]

y − 5 = [1/5](x + 3)

Find the equation of this line in slope intercept form.

- y − 5 = [1/5](x + 3)
- 5(y − 5) = [ [1/5](x + 3) ]5
- 5y − 25 = 1(x + 3)
- 5y − 25 = x + 3
- 5y − 25 + 25 = x + 3 + 25
- 5y = x + 28
- y = [(x + 28)/5]

y + 10 = [7/30](x − 2)

Find the equation of this line in slope intercept form.

- y + 10 = [7/30](x − 2)
- 30(y + 10) = [ [7/30](x − 2) ]30
- 30y + 300 = 7(x − 2)
- 30y + 300 = 7x − 14
- 30y = 7x − 314
- y = [(7x − 314)/30]
- y = [7/30]x − [314/30]

Find the equation of this line in slope intercept form.

- y + 12 = [5/6](x + 9)
- 6(y + 12) = [ [5/6](x + 9)]6
- 6y + 72 = 5(x + 9)
- 6y + 72 = 5x + 45
- 6y = 5x − 27

- m = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - m = [(5 − 9)/(2 − 7)]
- m = [( − 4)/( − 5)]
- m = [4/5]
- y − y
_{1}= m(x − x_{1})

- m = [(y
_{2}− y_{1})/(x_{2}− x_{1})] - m = [( − 11 − 3)/( − 7 − ( − 1))]
- m = [( − 14)/( − 6)]
- m = [14/6] = [7/3]
- y − y
_{1}= m(x − x_{1}) - y − 3 = [7/3][x − ( − 1)]

y = − [1/4]x − 3

This line also passes through the point ( - 5,7)

Find the equation of this line in slop intercept form.

- y = mx + b

parallel lines have the same slope

m = − [1/4]b = ? - 7 = − [1/4]( − 5) + b
- 7 = [5/4] + b
- 7 = 1[1/4] + b
- 5[3/4] = b

y = − [3/7]x + 6

This line also passes through the point ( - 4,2)

Find the equation of this line in slop intercept form.

- y = mx + b

parallel lines have the same slope

m = − [3/7]b = ? - 2 = − [3/7]( − 4) + b
- 2 = [12/7] + b
- 2 = 1[5/7] + b
- [2/7] = b

3x − 6y = 18

4x − 2y = 12

parallel, perpendicular, or neither?

- 3x − 6y = 18
- − 6y = 18 − 3x
- y = [(18 − 3x)/( − 6)]
- y = [18/( − 6)] − [3x/( − 6)]
- y = − 3 + [x/2]m = [1/2]
- 4x − 2y = 12
- − 2y = − 4x + 12
- y = [( − 4x + 12)/( − 2)]
- y = 2x − 6m = 2
- ([1/2]) ≠ − 1
- ([1/2]) ≠ (2)

5x − 2y = 10

2x + 12y = 24

parallel, perpendicular, or neither?

- 5x − 2y = 10
- − 2y = − 5x + 10
- y = [( − 5x)/( − 2)] + [10/( − 2)]
- y = [5/2]x − 5m = [5/2]
- 2x + 12y = 24
- 12y = − 2x + 24
- y = [( − 2x)/12] + [24/12]
- y = − [x/6] + 2m = − [1/6]
- (−[1/6])([5/2]) ≠ − 1
- (−[1/6]) ≠ ([5/2])

- y = mx + b

m = [1/5]

negative reciprocal = - 5 - − 3 = − 5( − 4) + b
- − 3 = 20 + b
- − 23 = b

- y = mx + b

m = − [7/10]

negative reciprocal = [10/7] - 4 = [10/7](5) + b
- 4 = [50/7] + b
- 4 = 7[1/7] + b
- - 3[1/7] = b

- 2x − 4y = 8
- − 4y = − 2x + 8
- y = [( − 2x + 8)/( − 4)]
- y = [1/2]x − 2
- m = [1/2]

slope of perpendicular line = - 2 - x − 5y = 20
- − 5y = − x + 20
- y = [( − x + 20)/( − 5)]
- y = [1/5]x − 4b = − 4

- 3x − 7y = 42
- − 7y = − 3x + 42
- y = [( − 3x + 42)/( − 7)]
- y = [3/7]x − 6

negative reciprocal = − [7/3] - 6x + 3y = 30
- 3y = − 6x + 30
- y = [( − 6x + 30)/3]
- y = − 2x + 10

b = 10

y = − [2/5]x − 6

This line also passes through the point ( - 10,12)

Find the equation of this line in slop intercept form.

- y = mx + b

parallel lines have the same slope

m = − [2/5]b = ? - 12 = − [2/5]( − 10) + b
- 12 = [20/5] + b
- 12 = 4 + b
- 16 = b

- m = − [2/3]

negative reciprocal = [3/2] - − 4 = [3/2]( − 6) + b
- − 4 = [( − 18)/2] + b
- − 4 = − 9 + b
- 5 = b

*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

### Linear Equations in Two Variables

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
- Objectives
- Linear Equations in Two Variables
- Point-Slope Form
- Substitute in the Point and the Slope
- Parallel Lines: Two Lines with the Same Slope
- Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
- Perpendicular Lines: Product of Slopes is -1
- Example 1
- Example 2
- Example 3
- Example 4
- Example 5
- Example 6

- Intro 0:00
- Objectives 0:13
- Linear Equations in Two Variables 0:36
- Point-Slope Form
- Substitute in the Point and the Slope
- Parallel Lines: Two Lines with the Same Slope
- Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
- Perpendicular Lines: Product of Slopes is -1
- Example 1 6:02
- Example 2 7:50
- Example 3 10:49
- Example 4 13:26
- Example 5 15:30
- Example 6 17:43

### Algebra 1 Online Course

### Transcription: Linear Equations in Two Variables

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

*In this lesson we are going to continue on with our linear equations and look at how we can build these linear equations for ourselves.*0002

*Specifically, some of things that we will look at is we will look at the point slope form on a line.*0014

*It will help us when building those equations.*0020

*For some of our examples, we will also have to know a little bit more about parallel and perpendicular lines.*0023

*Watch out for my explanation on what these are and how you can tell if two lines are parallel or perpendicular.*0029

*Earlier, we looked at two forms of a line.*0038

*We looked at standard form and we looked at slope intercept form.*0041

*Remember both of these forms are important because they were good for graphing.*0045

*When we want to build our own line, sometimes we can do this as long as we have enough information.*0057

*To form that we use for building our line is known as point slope form.*0064

*The reason why it is called that is because those are the two bits of information that we need.*0068

*We need to know at least one point on a line and we need to know what the slope of that line is.*0072

*Once we have both those bits of information and we can drop it into point slope form.*0077

*This form here requires a little bit of explanation.*0086

*You will see that there is an x and y that has some subscripts on it and that is where the point goes in terms of its x and y.*0090

*The slope is the n here so we can just put that in and then we also have a couple of other x’s and y’s which do not have subscripts.*0098

*With the ones that do not have any subscripts whatsoever, you will not be putting any values in for those.*0108

*Those simply stay in our equation.*0114

*One last thing to note is even though we will use point slope to actually build the equation on a line.*0116

*Usually you take point slope and end up converting it into slope intercept form because you are looking to do some other things without a line,*0122

*other than just than build it or maybe you want to eventually graph it.*0129

*It is good to go ahead and move it in to one of those other forms.*0132

*To build lines using point slope form, we want to substitute in our point and we will go ahead and substitute in our slope.*0143

*I already have point slope form, I will put it over here and I have a point and I have a slope.*0151

*Let us go ahead and just substitute these in and see how it works.*0157

*I will have y - -3 then the m represent the slope, I will put in ¾, x – xy.*0161

*Notice how the x and y which did not have any subscripts are still in there.*0176

*Once we have everything substitute in here, we want to go ahead and start rewriting this*0181

*either into standard form or slope intercept form so we can do some other stuff with it.*0186

*I’m going to go ahead and put this into slope intercept form by just getting y all by itself and maybe cleaning up a few other terms.*0190

*I'm subtracting a negative on the left side and that will be the same as adding 3.*0198

*I will go ahead and distribute my ¾ here.*0203

*I have ¾ x and -3, looking much better.*0206

*I will go ahead and subtract 3 from both sides.*0216

*Y = 3/4x – 3 – 3 is 6.*0219

*Our point slope gets our foot in the door so we can actually build the equation and then put it into a different form, maybe a slope intercept down here.*0227

*We can go ahead and graph it and do some other things with it.*0241

*Sometimes we will be given information about another line in order to build the one we want.*0247

*And information about that other line, we may know that it is parallel or perpendicular to the one we want.*0253

*What exactly does that mean?*0258

*We can say that two lines are parallel if they had exactly the same slope.*0261

*In my little picture here, you can see what that does.*0266

*You will end up with two lines and they usually have a little bit of a gap or space in between them, but they have exactly the same slope.*0269

*They are going in the same direction.*0275

*You might also have lines that are perpendicular.*0279

*With this one, they end up meeting at a right angle.*0282

*We do not talk much about angles in this course, another way that you can say two lines or perpendicular is*0286

*if their slopes are negative reciprocals of one another.*0291

*Let me show you what that means.*0295

*I suppose this blue line had a slope of 4/5, if the red one was perpendicular then I wanted to be the negative reciprocal of the blue line.*0296

*I made it negative and I flipped it over.*0312

*With perpendicular lines, they will always be different in sign, one will be positive and one will be negative and they will be reciprocal when flipped over.*0314

*Another way that you can test if two lines are perpendicular or not is actually you take both of their slopes and just multiply them together.*0325

*Two things could happen if you multiply them together and their slope is negative or their combined value is -1*0333

*then you will know for sure that they are perpendicular.*0342

*If you multiply them together and you get anything else other than -1 then you will know that they are not perpendicular.*0346

*It is a nice and easy task you can use to know how the two lines are related.*0352

*We will definitely know this as we get into more of those examples.*0358

*Let us start off with example 1, in this one we want to write the equation on a line using the given point and the slope.*0363

*Since the two bits of information that I need for point slope, I will just go ahead and nearly drop it into the formula.*0372

*y -y1 equals slope x - x1.*0378

*Let us put in the y value first, y - 3 equals my slope is -2/3x - -6.*0386

*You will know the formula has a minus sign in there since the x value is a negative, go ahead and put that negative sign in there as well.*0399

*Now all we got to do is clean it up a little bit and maybe turned it into a different form.*0408

*I will see what we can do.*0412

*y – 3 = -2/3, when I subtract the negative that is the same as addition.*0413

* x + 6 and I think I will go ahead and distribute this -2/3 in there.*0422

*-2/3x – 12 ÷ 3 and a -12 ÷ 3 = -4, it looks pretty good.*0430

*Let us add 3 to both sides and now we have that same line written into slope intercept form.*0449

*Again we use point slope form to go ahead and create the equation of the line and then probably put it into some other form.*0462

*This one is a little different, in this one we simply want to determine whether the two lines are parallel*0472

*or maybe the perpendicular or maybe they do not fall into either of those two cases.*0477

*With these ones, notice how many of them are written not in one of the forms that we have covered.*0482

*In other words they are not in slope intercept form but the second one is in standard form but we have to be able to figure out what their slope is.*0489

* A way to determine what the slope of the line is to go ahead and rewrite it into our slope intercept form and we can just read it out of the equation.*0501

*Let us do that first before we actually compare what these slopes are.*0509

*Starting with this first one here, I can move the 2 to the other side by subtracting 2 so y =5x – 2.*0512

*Let us see what the second one, let us go ahead and move the x to the other side, -x – 15 and then we will divide it by 5.*0521

*Here is our first line, here is our second one.*0539

*Now they are both in slope intercept form we can say that the first one has a slope of 5*0545

*since it is right next to x and the other one has a slope of -1/5.*0550

*Looking at the two slopes, they are definitely not the same, they are not parallel.*0555

*It looks like one is positive and one is negative and they are reciprocals.*0562

*I think these two are perpendicular.*0567

*If you want to test out feel free to just take the two slopes and multiply them together and see that when you do this you get a -1.*0570

*You will know that these two lines are perpendicular.*0579

*Let us try this process with the next pair of lines.*0590

*I will begin by just putting this into slope intercept form.*0595

*This one is almost in slope intercept form.*0599

*I just have to reverse the y and putting it first.*0602

*This one I will go ahead and move the 3x to the other side, so now I have my two lines right here.*0606

*y = 2x + 1 and y = -3x + 4, in this form, I can read off what both of their slopes are.*0616

*Looking at their slopes I can see that they are not the same, they are definitely not parallel.*0627

*One is positive and one is negative, but they are not reciprocal so they are not perpendicular either.*0633

*They are not parallel or not perpendicular, I will put this in the neither categories.*0638

*They are just two lines hanging around on the graph.*0643

*Let us try that same process with another pair of lines.*0650

*Let us see if we can find a couple of that which actually are parallel.*0652

*We will begin by putting these into a better form so we can read off that slope.*0656

*I'm moving the 4x to the other side, this one the y is still not completely all by itself.*0662

*Let us go ahead and divide everything through by 2, now we have one of our lines.*0668

*The second one let us go ahead and rearrange things.*0678

*The y is on the left and let us get it completely isolated by multiplying everything through by -1.*0683

*Now we have both of our lines.*0693

*Now that they are in this form, the slope of the first one is -2 and the slope of the second one is -2.*0702

*I can see that they are exactly the same and we will call these pair of lines parallel.*0709

*Onto one more pair of lines, let us see what they are, parallel, perpendicular, or neither, we will find out.*0719

*I need to get my y isolated I will start off by moving the 4x to the other side.*0726

*I will divide both sides by 3, y =4/3x +2.*0734

*The second one, let us go ahead and move the 2 over.*0745

*I have 3x + 2 and I will divide everything by the 4, ¾ x + ½.*0749

*Here is equation 1 and here is equation 2.*0760

*If you look at them in this form, we can pick out what each of their slopes are, I have 4/3 and 3/4.*0767

*These ones are pretty close, they are definitely reciprocals of one another since you would take 4/3 and flip it over and get ¾.*0775

*When those have been both positive so we can not say that these are perpendicular.*0783

*Even though they are reciprocals they are not negative reciprocals of one another.*0788

*They are definitely so they are not parallel.*0792

*Even though they are close, they do end up in the neither category.*0795

*They are not parallel or not perpendicular, they are simply neither.*0799

*They are just two lines.*0802

*With this one we want to write the equation of a line that happens to be parallel to the given line and also goes through the given point.*0809

*This one is a little different, notice how we need to know what the slope of the line is*0818

*and we need to know what the point is but they have not quite given us what the slope is.*0822

*We are going to have to extract that out of this other line that they have given us by knowing that it is parallel to the one we want to build.*0827

*Let us hunt down what the slope of this other line is.*0834

*We will do that by dividing everything through by 4.*0837

*It looks good so y = ¼ x + 5.*0845

*Let us see, I know that the slope is ¼.*0850

*Now that I have a point, I have a slope, I can use point slope form and build our line.*0855

*Y - -3 = ¼ x – xy and now let us clean it up and see what line we have.*0861

*y + 3 = ¼ x, ¼ of 2 would be -1/2.*0872

*Now we will subtract 3 from both sides.*0890

*y = ¼ x – ½ - 3, get a common denominator over there.*0893

*Our line is y = ¼ x – 7/2.*0908

*The way we built this line, I know for sure that it goes through the given point 2, -3.*0917

*I already explored what the slope of the other line was so I know it has a slope of 1/4.*0923

*Let us try this one, write the equation of a line which is perpendicular to the given line and it goes through the given point.*0932

*Similar to the other one, but it have not given us the slope directly.*0939

*It just told us it is perpendicular to this other line over here.*0943

*Let us figure out what slope is, y equals, I will move the 8 to the other side, -8x + 3.*0948

*The slope of this line is -8 which is good but that is not exactly the slope we want to use.*0958

*Our line is perpendicular to this line, so we need to use the slope -1, 1/8.*0966

*That way it is the negative reciprocal of the other.*0979

*Now we will drop it into our formula.*0984

*Let us go ahead and put in our y value.*0994

*We have our slope and our x value.*0997

*We will clean it up and see what line we have.*1002

*y - 8 = 1/8 - 4/8, my x in there.*1005

*y – 8, 1/8x – ½ and we will go ahead and 8 to both sides, find a common denominator and combine the last of our like terms.*1017

*1/8x + 15/2.*1044

*We can see that its slope is the negative reciprocal of the other line.*1053

*I know it is perpendicular and the way we build that it, for sure it goes through the point 4/8.*1057

*This last one is a little bit more of the word problem but you can see how we can still use these techniques in order to build the equation of the line.*1066

*We want to build a line that represents the following problem.*1074

*It cost a $20 flat fee to rent a drill + $2 every day starting with the first day.*1077

*Let x represent the number of days that we rent this drill and y represent the charge to the users.*1085

*How much will we end up willing to get from them?*1091

*When we are all done, let us see if we can write this line in a slope intercept form.*1095

*Using the techniques that we know about so far, we need to figure out what our slope is and what point this will go through.*1100

*Let us see, one thing that we can think of is the slope being what changes or your variable costs.*1108

*The $20 that you have to pay up front is not a variable.*1120

*It is going to be there no matter what, what does change is a $2 every single day.*1123

*The variable cost, we can think of that as our slope.*1134

*Let us see what we can do with that.*1144

*What happens with that flat cost?*1146

*We will be charged of that no matter what, that is like our y intercept.*1151

*If this one, we can just drop in our variable cost and we can drop in our flat fee.*1158

*What we are left here is the equation of the line but it represents how much we end up charging the person*1177

*and we can see this by substituting some values.*1183

*Suppose they rent this drill for one day, it cost $2 for that day + $20 flat fee, it cost them $22.*1186

*If they rent it for two days, we can put that in for x, $4 for the variable + $20 flat fee, $24.*1196

*You can go on and on figuring out how much it cost for each of the different days.*1206

*But in the end, this formula right here would represent how much we need to charge them.*1211

*Just simply plug in the number of days for x and y would give you the cost.*1215

*You can see that building a line is not so bad and using point slope form is handy in that entire process.*1221

*If you do use point slope form, you need to know the slope of the line and you need to know a point on that line.*1228

*Thanks for watching www.educator.com.*1234

1 answer

Last reply by: sherman boey

Tue Aug 12, 2014 11:13 AM

Post by sherman boey on August 12, 2014

example 5 is wrong.. y =8 not -8

1 answer

Last reply by: Professor Eric Smith

Sun Jul 6, 2014 2:18 PM

Post by patrick guerin on June 24, 2014

Thank you for the lecture!

0 answers

Post by Farhat Muruwat on March 21, 2014

can someone please fix the answers to the questions on the practice questions.

0 answers

Post by Professor Eric Smith on December 10, 2013

The key you want to recognize for many of these problems is that to build a line using point-slope form we need to know a slope, and a point on the line. In the problems we are usually given just enough information to find these things, but may have to use other formulas to do it. Let me know if that helps out. :^D

0 answers

Post by Araksya Fernandes on December 10, 2013

is there a little simple way to explain similar examples.