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Robert McKeown: Hello, and welcome to ALEKS

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walkthrough video number four. My name is Robert

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J McKeown, and I'm very happy that you decided

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to check out my video today we're going to be

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covering lines and systems. Now I'm not going to

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go through every single concept and ALEKS topic

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for that would just take too long. And I want

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you to try and do it yourself, because that's

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how you're going to learn. Today's topic is

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relatively straightforward. One thing you'll

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hopefully appreciate from this activity is that

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linear equations are great, because with a

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linear equation, there's typically one solution,

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which is really, really handy, such as x is

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equal to four. That's really nice. Remember,

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last time in the last video, we had quadratics,

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where we could have two solutions. So linear

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equations are really great to work with. And

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we're also going to be working with graphs. And

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so this is our first sort of pass at graphing. A

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few things to remember one, understanding how to

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read graphs and how to create graphs is very,

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very important for you, as a young person

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learning to be an economist and learn to be a,

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what we call a white collar worker, and a

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knowledge worker. graphs are a way of

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representing sophisticated or complicated

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mathematical and statistical information to a

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wider audience. I make it makes it more

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accessible to people that maybe don't know that

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much about stats or math. But it's all and it's

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also very succinct, and can get that message

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across very quickly to people who do know quite

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a bit about statistics and math. And so it's

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going to be extremely important for your career,

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whether you're working in the private sector,

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wherever even if you're working in, you know,

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charities, the public sector, if you're working

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academics, being able to present information in

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a easy to understand and accessible way is a

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great way to get promoted, and to have a really

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great career. So I hope you've got your pencil,

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you've got your paper, and maybe you've got

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another tablet, if you want to write on that

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tablet, instead of using the pen and paper.

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Either way, I hope you're ready to go. So let's

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jump into the questions. Our first question says

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find an equation for the line that passes

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through the points two comma negative four, and

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negative four comma four. So if you're new to

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this, these are Cartesian coordinates. And the

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first value is the x coordinate. And the second

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value is the y coordinate. So if I draw a little

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diagram, I've got a coordinate at here's the x

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axis, here's the y axis, I have a point here at

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two x is equal to two. And when x is equal to

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two, y is equal to negative four. So I've got a

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point here, that's two and negative four. Now,

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we might want to say this is our first point,

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and this will be our second point. So I'll put a

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subscript of two to denote that it's the second

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observation or the second point, and here x is

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equal to negative four. And when x is equal to

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negative four, y is equal to four.

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And now we've identified the point to negative

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four. Now it's a line. This is an equation for

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the line the past the points described. So I

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could draw a line like that and it goes off

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forever in both directions. Now the question is

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asking me to find an equation for this line.

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That equation for a line

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always has the form y is equal to the intercept

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A plus the slope times the x variable. So A is

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the intercept, or I should be careful, that's

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the y intercept. m is the slope. And x is the x

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axis value. And so the question is asking me to

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find, essentially this point right here,

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I want to find that point, the equation to find

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that point is one you're probably familiar with,

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which says that the slope of a straight line, or

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the slope of any line for that matter, is equal

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to rise over run. And in our case, rise over run

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is equal to the change, or maybe the distance

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from y two, two y one divided by the distance

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between x two and x one. And if we do that,

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we're going to find the slope. So let's go ahead

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and find the slope of this line. I've got m is

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equal to four minus minus four, that's y two

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minus y one divided by negative four, minus two.

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And that's going to give me eight over negative

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six, which I could simplify to negative four

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over three. And so the equation, the equation,

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this line is going to have a slope that's equal

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to negative four over three. So I have y is

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equal to the y axis intercept, minus four over

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three times x. Now, how am I going to find the y

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axis intercept? Well, I'm going to go back to

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our rise over run equation. And I'm going to use

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these coordinates as my x one and y one

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coordinates. So I'm going to have a negative

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negative four in the numerator, oops, and a to

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negative two and the denominator. And if I want

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to find the y intercept, the y intercept occurs

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when when x is equal to zero when x is equal to

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zero, so I'm going to let x two equal to zero.

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And I'm going to let this whole thing be equal

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to negative four over three, and I'm going to

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solve for y two, I'm going to solve for y two.

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I've got negative four over three is equal to y

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two plus four over negative two. If I multiply

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both sides by negative two, I get eight over

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three is equal to y two plus four, I get y two

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is equal to eight over three minus 458 over

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three, that's like minus 12 over three. It's

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equal to y two and I get y two is equal to

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negative four over three. Now that looks a

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little silly. Because I have the same value for

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the slope that I have for the y axis intercept,

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this implies that the equation

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of this line is equal to negative four over

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three minus four over three times x. Right? And

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that's the same form

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that we started out with. Here I am on ALEKS.

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I'm going to put our answer and and see if we

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have the right answer. So I've got a y, y is

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equal to negative four over three times negative

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for over three, multiplied by x. And although it

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looked kind of funny, it was the right answer.

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So we got the right answer. And that's the

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methodology you want to be able to use. So we're

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actually able to use the same equation or

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equation for the slope of the line twice.

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Looking at our next question, we want to graph

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the solution to a system of inequalities. This

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is pretty straightforward, but I found it very

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easy to make a mistake when I was working,

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especially working through ALEKS. So let's take

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a look at the first equation, I want to graph

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the first equation we have y is greater than two

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x minus three. So there's a couple ways to

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approach this, I would recommend finding two

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points in this inequality. And starting there,

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so I'm going to say well let x equals zero. If x

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equals zero, y becomes two times zero, minus

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three and y is equal to minus three. Now I'm

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going to go over to ALEKS. And I'm going to make

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a little mark. So we've got x is equal to zero,

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y is equal to minus three. And I made, it's kind

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of hard to see, but I made a little a little

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tiny blue x at the coordinates. Remember our de

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Cartesian coordinates at zero, minus three. Now,

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what if I say, well let x equal one. Now I've

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got and maybe, you know, I should probably be

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aware, I like to put little equalities there.

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But of course, this is an inequality. So maybe

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I'll keep the inequality live. So if we let x

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equal one, y is going to be larger than two

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times one, minus three, so y is going to be

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larger than negative one. Now we'll go back to

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ALEKS. I'll click on the little pencil diagram.

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And I'll make a point. Maybe I should have done

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it over here. First, I'm looking for the

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coordinate one, negative one, one, x is equal to

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one and y is equal to negative one. Now that

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I've got those little coordinates marked, I'm

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going to just press on that line icon, and

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essentially tap or click on those two little

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axes. And now I've got a line going up and has a

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slope that's equal to two, you should be able to

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say that the slope here is equal to two. Now y

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has to be larger, y has to be larger, or above

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and to the left of this line, and I'm going to

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click on this icon right here that I've got

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highlighted, and he should be able to see with a

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little hand pointer. Now if I click on that, and

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I click on this space up here, y can take on any

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of the values of that shaded area. Now before I

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rush off, when I start working on the second

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inequality, notice that y is strictly greater

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than to x minus three. So I'm going to click on

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this little icon here, you can see that it's

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highlighted and that light blue color, and it's

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got an arrow pointing in both directions. I'm

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going to click on that. And I'm going to click

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on the line. And now notice that the line has

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become

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the line has become dotted. And that means y

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cannot take on a value equal to two x minus

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three. So that's just a little a little

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technical, a little bit of detail and definitely

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detail you want to get correct when you're

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working with ALEKS. So that you get rewarded for

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everything that you do now. Now we're ready to

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tackle the second inequality and we've got y is

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greater than or equal to negative two x minus

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seven. And I'm going to do the same thing, well,

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let's let x equal to zero. That's easy

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calculation. I like using that as one of the

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points, then we get y is greater than or equal

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to negative seven. And I've got the point zero,

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negative seven. What if we let x be equal to

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one. Now we've got y is greater than or equal to

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negative two, excuse me, times one, negative

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seven, now y is greater than or equal to

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negative nine. And so we've got the coordinate

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one, negative nine. Now I'm going to go and I'm

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going to use the little pencil icon like I did

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last time. And I'm going to look for zero

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negative seven, which is down there. And I'm

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going to look for one and nine, which is right

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down there. Now, I'm going to put cross the line

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icon, and I'm going to click on those two little

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axes that I drew on the diagram. It's already

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been shaded in for us. Why can take on it can be

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equal to negative two x minus seven with our

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second equation, and so why don't we see if I've

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done this correctly, I'll click click the check

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button. We'll see if we've got the solution. And

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we do so as long as y is in this blue, light

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blue shaded area here