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Is the system of linear equations below

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dependent or independent, and they gave us

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two equations right here.

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Before I tackle this specific problem,

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let's just do a little bit of review of what

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dependent or independent means.

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And actually, I'll compare that to

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consistent and inconsistent.

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So, just to start off with, if we're dealing

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with systems of linear equations in two dimensions,

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there's only three possibilities

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that the lines, or the equations,

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can have relative to each other. So let me

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draw the three possibilities. Let me draw

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three coordinate axes. So that's my first x-axis

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and y-axis, so x and y. Let me draw another

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one. And that is x, and that is y. Let me draw

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one more, because there's only three possibilities

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in two dimensions, x and y, if we're dealing with

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linear equations, x and y. So you can have

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the situation where the lines just intersect in

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one point, so you can have one line like that,

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and maybe the other line does something

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like that, and they intersect at one point.

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You could have the situation where the two

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lines are parallel. So you could have the situation

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actually let me draw it over here, where

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you have one line that goes like that, and

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the other line has the same slope, but

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it's shifted, it has a different y-intercept,

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so maybe it looks like this, and you have no points

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of intersection. And then you could have the

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situation where they're actually the same line,

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so that both lines have the same slope

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and the same y-intercept. So really, they are

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the same line, they intersect on infinite number

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of points. Every point on either of those lines

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is also a point on the other line. So just to give

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you a little bit of the terminology here, and

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we learned this in the last video, this type of

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system, where they don't intersect,

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where you have no solutions, this is an

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inconsistent system.

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And by definition, or I guess just taking the

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opposite of inconsistent, both of these

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would be considered consistent. Both of these

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are consistent.

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But then, within consistent, there's obviously

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a difference, here we only have one

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solution, these are two different lines

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that intersect at one place. And here,

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they're essentially the same exact line.

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And so we differentiate between these two

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scenarios by calling this one over here

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independent, and this one over here dependent.

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So independent, both lines are doing their own

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thing, they're not dependent on each other,

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they're not the same line. They will intersect

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at one place. Dependent, they're

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the exact same line, any point that satisfies

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one line will satisfy the other.

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Any point that satisfies one equation,

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will satisfy the other. So with that said,

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let's see if this system of linear equations

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right here is dependent or independent.

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So they're kinda having us assume that

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it's going to be consistent. That we're either

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going to intersect at one place, or we're

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going to intersect at an infinite number

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of places. And the easiest way to do this,

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we already have this second equation here,

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it's already in slope-intercept form,

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we know the slope is -2, the y-intercept is 8.

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Let's put this first equation up here in

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slope-intercept form and see if it has

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a different slope or a different intercept,

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or maybe it's the same line. So we have

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4x plus 2y is equal to 16.

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We can subtract 4x from both sides,

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what we want to do is isolate the y on the

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left hand side. So let's subtract 4x from

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both sides. The left hand side we are

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just left with a 2y, and then the right hand side,

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we have a -4x plus 16. I

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just wrote the -4 in front of the 16

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just so that we have it in the traditional

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slope-intercept form. And now we can

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divide both sides of this equation by 2,

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so that we can isolate the y on the left-hand side.

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Divide both sides by 2, we are left with

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y is equal to -4 divided by 2 is -2x

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plus 16 over 2, plus 8. So all I did is

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algebraically manipulated this top equation

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up here, and when I did that, when I solved

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essentially for y, I got this right over here,

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which is the exact same thing as the

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second equation. We have the exact same

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slope, -2, -2, and we have the exact

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same y-intercept, 8 and 8. If I were

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to graph these equations, that's my

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x axis, and that is my y-axis,

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both of them have a y-intercept at 8

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and have a slope of -2.

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So they look something, I'm just drawing

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an approximation of it, but they would look

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something like that. So maybe this is the graph of

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this equation right here, this first equation.

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And then the second equation will be

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the exact same graph, it has the exact

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same y-intercept and the exact same slope.

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So clearly these two lines are dependent.

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Dependent. They have an infinite number

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of points that are common to both of them

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because they're the same line.