In the series: Note 2.

Subject: Linear equations

Version: 0.3

By: Albert van der Sel

Doc. Number: Note 2.

For who: for absolute beginners.

Remark: Please refresh the page to see any updates.

Status: Ready.

So really..., my emphasis is on "rather

So, I am really not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

Contents:

1. Positions in a Coordinate Sytem for the XY "plane".

2. Linear equations "y = ax + b" (they represent "lines" in a Coordinate system).

3. The equivalent "ax + by = c" notation.

4. Solving linear equations (finding the intersection of 2 lines).

5. The "slope" of a line.

6. Finding the equation of a line, if you have two points.

7. An important property of the slopes of 2 perpendicular lines: a

8. Optional: vector representation of lines.

Section 8 is optional. Actually, one should first have studied some basic material on "vectors".

No problem. Then just skip this one. You can always return to it at a later moment.

Long time ago, mathematicians deviced a rather nifty method to accomplish that purpose.

They designed a "grid", with a well-defined point of reference. This special point, is often called

"the Origin". Next, two perpendicular "axes" are drawn, which intersect exactly at the Origin.

The horizontal axis, is called the "x-axis". The vertical axis, is called the "y-axis".

So how do we pinpoint "points" in that Coordinate system?

Take a look at figure 1.

Figure 1. Coordinate system (in the plane).

From the Origin (

Or more precise, if you take 3 steps to the right, you are at x=3

along the "y direction" at all.

In the next example, we will move in both the x- and y directions.

From the Origin (

If you now, from that position (x=3), take two steps up in along the y-direction, then you

arrive at x=3 and y=2. The way how often mathematicians write that down is: we are at point (3,2).

Do you see the point (3,2) in figure 1 above?

We do not only have positive numbers like 1, 3, 12, 1000 etc.., but negative numbers as well.

Those are numbers like, for example, -1, -3, -12, -1000 etc..

If you are not "comfortable" with negative numbers, then in this link (section 3),

I try to explain it a bit.

In the Coordinate system, if we take steps

However, if taking steps

than using different numbers. You know, the positive ones

But it really makes sense to denote "moving to the left" as saying that we go to x=

It makes it completely "symmetrical". For example, from the Origin, if you take 5 steps to the right,

you go to x=5. But, if you, from the Origin, take 5 steps to the left, you end up in x=

In such a way, 5 steps to the right, or 5 steps to the left (from "O"), have the same absolute distance.

In fact, only this way we have a truly usable Coordinate system.

As a last example, note the point (-3, -3) in figure 1. It means: x=

If you want to "pinpoint" the position, you may say: it's the point where x=

But it's much easier to pinpoint it by using the notation (-3,-3).

In words: (-3,-3) means: from the Origin, I take 3 steps

I take 3 steps

I will explain the equation in a minute. But first a few words on

About the "format" of such equation:

If you look at such equation "

along the x-axis, and not the multiplication symbol. That symbol is usually simply left out.

So, y=ax+b, where a and b are some constants (like "2" and "3"). It expresses how "y" depends on "x", that is,

what the values of "y" are, when we vary "x". It turns out to be a straight line. We will explain that below.

Now about the meaning of such equation.

If you see such a formula for the very first time, I can understand that you think

We already have seen the Coordinate system with the x- and y axes. You can define, or pinpoint, "points" in the plane,

by specifying the exact "x" and "y" for such a point, like for example the point (2,4).

Let's take a look at a specific example, like "y=2x".

And.. let's see what happens with the value of "y" if we fill in a couple of "x values". Here we go:

Table 1: calculate "y" for a number of x values.

value of x: | value of y: (which is 2x) |

x= -2, then | y= -4 (because 2 x -2 = -4) |

x= -1, then | y= -2 (because 2 x -1 = -2) |

x= 0, then | y= 0 (because 2 x 0 = 0) |

x= 1, then | y= 2 (because 2 x 1 = 2) |

x= 2, then | y= 4 (because 2 x 2 = 4 |

Figure 2. The functions y=2x (red) and y=2x + 3 (blue)

In figure 2, we simply have plotted the points from Table 1, into the Coordinate system.

If we connect those dots, we simply get a line ! See the red line in figure 2.

We may conclude that if we put the equation "y=2x" into the Coordinate system (by calculating y values,

for a certain number of x values), we then find a line. A line is also often called "a linear equation".

So, an equation like "y = ax", where a is just some constant number, can be represented by a line.

You could perform a similar action for other equations like for example "y=5x" etc..

By the way: note that for drawing a line, you actually need only two points.

You may ofcourse calculate for a larger number of "x-values", what then the resulting "y-values"

will be, just like we did in Table 1. But actually, doing so for 2 points only, is enough.

So, we analyzed the equation "y=2x" and saw how we could plot points in the Coordinate system.

That example stands model for any sort of such equation "y=ax", where "a" is some number.

However, a general linear equation is this:

So, what's the story with "b"?

Well, in figure 2 I have placed the example line "y= 2x +3".

Note that

So, it's always 3 above "y= 2x" with respect to the y values.

Does that make sense? Take a look again at both examples:

y= 2x

y =2x +

Yes, the only difference is the '+3', which means that for all x on the x-axis, y is always '+3' larger

compared to the equation "y=2x".

In figure 2, notice the blue line. It's the line "y=2x +3".

Question: could you create a similar table like Table 1, but this time for "y=2x +3"?

By the way, the answer can be found in figure 2 as well. Note the column on the right side.

is that you

For example, you do not need to draw a line first in a Coordinate system. Say x=30, then y=7x30 + 5 = 215.

And that holds for any "x". Just do the calculation, and you know the value of the corresponding "y".

For each different "a" and "b", we have a different line in the Coordinate system.

However, what happens if "ax=0"? Then we simply have "y=b".

This means that,

So we could have the line "y=5", or "y=-7" etc.. These lines are parallel to the x-axis, since y remains constant.

Here are some examples:

Figure 3. Some examples of the equation "y=b" (thus y is constant).

Here, the number "a" is a negative number. Nothing changes fundamentally, however, if you are really

new in this business, you just need an example of such a case.

Let's draw the line, with the help of a table, similar to Table 1.

But..., you need to know a certain fact beforehand. Maybe you already know this, or not, in which case in cannot hurt.

Here is what I mean:

a positive number X a positive number = a positive number. For example 5 X 5 = 25

a positive number X a negative number = a negative number. For example 5 X -5 = -25 (note the "-" signs).

a negative number X a negative number = a positive number. For example -5 X -5 = 25.

So, for example 5 x 5 maybe seen as "Do 5 times, 5 steps to the right from the Origin",

which will put you at x=25.

For example, 5 x

which will put you at x=

For example,

which will put you at x= 25.

Indeed, a "negative number" times a "negative number" is positive.

Ok, it's time to draw the line:

Table 2: calculate "y" (y = -3x + 2) for a number of x values.

value of x: | value of y: (which is -3x + 2) |

x= -2, then | y= 8 (because -3 x -2 + 2 = 6 + 2 = 8) |

x= -1, then | y= 5 (because -3 x -1 + 2 = 3 + 2 =5) |

x= 0, then | y= 2 (because -3 x 0 + 2 = 0 + 2 = 2) |

x= 1, then | y= -1 (because -3 x 1 + 2 = -3 + 2 = -1) |

x= 2, then | y= -4 (because -3 x 2 + 2 = -6 + 2 = -4 |

Since we have two or more points of this line, we can draw it now:

Figure 4. The line y = -3x + 2.

We have seen the functions of the form "y=ax+b". There is however a different

way to write them, but both ways are fully equivalent.

mathematical

Let's take a look at a few examples:

Suppose we have:

y = 2x + 3

Then how to rewrite that to the other format?

It's really easy. On both sides of the "=" symbol, we may add or subtract any number, or term,

and it makes not difference. As long as we do it at both sides.

You may even multiply all terms on both sides of the "=" symbol, with the same number.

So, if we add -2x to both sides, we have:

-2x + y = 2x + 3 -2x

And that will resolve to:

-2x + y = 3.

Yes, it looks different, but it's the same as "y=2x+3".

Suppose we have:

y = -4x + 3

Let's rewrite it to the other format.

On both sides, I may add +4x. So then the equation becomes:

4x + y = -4x + 3 + 4x

On the left, the "+4x" and "-4x" cancel out, so we have:

4x + y = 3

You can always rewrite such an equation to the older format "y = ax + b".

There are some advantages of the format "ax + by = c", compared to the traditional equation.

You will see them in later notes. One such advantage is that the "vector" perpendicular to

the to line represented by "ax + by = c", is the vector

You will see that in later notes.

between x and y in the entire Coordinate system.

Solving an equation can mean that you can calculate a

For example:

What is the value of "y" if x=3 for the equation y=5x +2 ?

Answer: y= 5 x 3 + 2 = 15 + 2 = 17.

However, what is usually meant by solving, is finding "special points".

Any line (except the horizontal and vertical lines), will "intersect" with the x-axis and y-axis.

Finding those (x,y) points, can be of great use:

At such a specific point, thus where the line crosses the y-axis, then we know for sure that "x=0".

Take a look at the figures above.

From those examples, you can really see that

Thus for that specific point:

y= ax +b and with x=0, we have y= a x 0 + b = b

For example: the line y=2x + 3 crosses the y-axis at: x=0 and y=3, or the point (0,3).

If a line crosses the x-axis, then at that specific point,

Take a look at the figures above. At the x-axis, the height of y is zero, so y=0 at that point.

Thus:

y= ax +b and if y=0, then ax +b =0, thus x=- b/a

Note:

If we have: ax + b = 0

Then we may add "-b" to the left side- and right side of the "=" sign. Thus:

ax + b = 0 =>

ax + b -b = -b =>

ax = -b =>

x= -b/a

So, for example, suppose we have the line y=2x +4. Where does it intersect the x-axis? At that point, we know for sure that y=0. So:

2x + 4 = 0 =>

2x + 4 -4 = -4 =>

2x = -4 =>

x= -4/2 = -2

So, this line intersects the x-axis at x=-2, or at the point (-2,0).

Two lines which are not parallel, will intersect at a certain point. Ofcourse, since the lines are straight,

there can

See figure 5 for an illustration.

Figure 5. Intersecting lines

One thing is for sure: where they intersect,

So, suppose we have:

y= -3x - 3

y= -x - 1

Since at that point y=y, we have:

-3x + 3 = -x - 1

Since an equation stays the same if on both sides of the "=", the same is added.

Let's add "x" to both sides:

-3x + 3 + x = -x -1 + x

thus:

-2x + 3 = -1

Add -3 to both sides:

thus:

-2x = -4

thus:

x = -4/-2

x = 2

So, at least the x-coordinate is known. It's x=2. Now you can use either of both equations to calculate

the y-coordinate. Let's use "y= -x - 1". Thus y = -2 - 1 = -3.

Thus we have found the point (2,-3), where both lines intersect.

rise moderately, of even slowly as "x" increases. It just depends on the "slope" of the line.

The figure below, illustrate some lines with a different "slope".

Figure 6. A few lines with a different slope.

-Please take a look at the line y=3x+3.

Here you see that if you take one "x step" to the right, then "y climbs" 3 steps.

Maybe you can also note the small triangle near that line, illustrating the effect.

-Please take a look at the line y=½x-2.

Here you may notice, that you must take "two x steps" to the right, before "y climbs" only one step.

This illustrates the "direction coefficient", or "slope" of a line.

Notice that the "a" in the equation "y=ax+b", fully determines the slope of a line.

For example, in the two examples above, we had a line with "a=3", which shows a line

which rises rather steeply.

But in the other example, we had a line with "a=½:, which shows a line

which only rises slowly.

It's important to see that in the equation "y=ax+b", the "a" immediately shows you the slope

of that line. For example, "y=12x+3" rises very steeply, since "a=12".

Note, that if "a" is a positive number, then we have a rising line (if you move to the right on the x-axis).

Also note, that if "a" is a negative number, then we have a declining line (if you move to the right on the x-axis).

Indeed, you may for example take a look at the line "y = -2x + 2", en then you will see that

it declines if you move to the right on the x-axis, but it rises from the perspective if you move to the left

on the x-axis.

The "slope" of lines will be explored further, when we will study the socalled "derivative" of functions.

in the number of steps, if you took one step of "x".

But it does not have to be one step of "x". You may also take for example 3 steps of "x", and see how

the number of "y steps" changed.

The ratio of

a constant slope.

Suppose from a certain line y=ax+b, you found that with one step of "x" to the right, then "y" climbs 4 steps.

Then the slope is:

slope= |
4 - 1 |
= 4 |

Note that you now also immediately have found the "a" of the equation "y=ax+b".

So, at least we now know that the equation is "y = 4x + b".

To make this a bit more general: take any increase of x as you like (denoted by Δx),

and determine the corresponding change in y (denoted by Δy), then:

slope= |
Δy -- Δx |

You may also say:

y = ax + b | = | slope * x + b | = |
Δy -- x Δx |
+ b |

if you know the slope of that line. Then, all that's left to do, is finding "b".

So, suppose we have the points (2,1) and (6,2). What is the equation of the line

which goes through both points?

From the coordinates of those 2 points, we see that If we move 4 steps the the right on the x-axis,

then y has climbed only one step.

Thus:

slope= |
1 - 4 |

At this point, we know that the equation must be "y = ¼ x + b".

We only need to use one of those two points, (2,1) or (6,2), and substitute those x,y values

into the equation "y = ¼ x + b". Suppose we use (2,1). Then:

1 = ¼ * 2 + b

Thus:

1 = ½ + b

Thus:

b = ½

So, our equation is: y = ¼ x + ½

Suppose you see a task, showing a table with x and y values, and they want you to

find the equation of the corresponding line.

Suppose you see a table like this:

x: |
y: |

1 |
4 |

2 |
6 |

3 |
8 |

4 |
10 |

This table shows 5 "x"'s and 5 corresponding "y"'s. So, you have 5 points (x,y), and you only

need two points of them, to get the equation of the line.

Just pick two points, and follow the procedure as shown above.

Note that "if you walk along the x values", the corresponding y's always increase with a fixed

value, compared to the former y value. Here, it is "2".

Ofcourse it works that way. The slope of the line is constant, and we know that the slope is:

Δy -- Δx |

The slope must be a constant for the whole line, and in this example, it's "2".

The proposition is, that if you would have two perpendicular lines:

f(x) = a

g(x) = a

then:

two randomly choosen lines, which are perpendicular. You ofcourse, can take any

other two perpendicular lines, if you want.

See the figure below, which illustrates two random perpendicular lines.

Figure 7. For two perpendicular lines, then "slope1 x slope2 = -1".

Here we have the lines:

f(x)=3/2x + 6.5

g(x)=-2/3x

It's easy to see that in this case:

Suppose we have f(x)=3x -2

Determine the line g(x) which is perpendicular to f(x), and also passes through the point (-3,2).

We immediately know the slope of g(x). Since:

3 x (slope of g(x)) = -1

It follows that the slope of g(x) = -1/3.

Thus: the linear formula of g(x) is:

Now, we only need to determine "b". That's easy, since g(x) must pass through (-3,2).

Thus:

2 = -1/3 x -3 + b => b=1.

Thus the answer is:

No problem. Then just skip this one. You can always return to it at a later moment.

For example, after you have studied "note 12" on "vector calculus".

(orthogonal) coordinate system (using an x- and y axis). Such a line can be written as "y = ax + b".

However, from "vector analysis", we know how we can represent vectors in some space (like a 2dimensional,

or 3dimensional space). If needed, study note 12 again (up to section 2.1 is quite enough).

Also, we know how to add two vectors, and we know of some other operations as well.

Take a look at figure 1 below:

Figure 1: two lines in R

We see two lines, with the familiar notations "y = 3/2 x + 2" (the red one), and

"y = -2x + 10" (the blue one).

We can also represent the lines in vector notation. Let's try this for the red line:

┌ x ┐ └ y ┘ |
= |
┌ -4 ┐ └ -3 ┘ |
+ | λ |
┌ 2 ┐ └ 3 ┘ |

What a difference. No not really. The vector notation above, can be explained. It says: any point (x,y)

on this line "can be reached" by starting with vector (-4,-3) and, then adding the vector (2,3), where

λ defines the "amount" of which (2,3) must be multiplied.

So, for example the point (0,3) clearly is a point on the red line. To get there, from the Origin,

we can write it like so:

┌ 0 ┐ └ 3 ┘ |
= |
┌ -4 ┐ └ -3 ┘ |
+ | 2 |
┌ 2 ┐ └ 3 ┘ |

Now, to reach any arbitrary point on the line, λ must take on specific values for those points.

So, for all points on the line, λ takes on values from the "real numbers" ℝ, like

for example 2, -2, 5, 100, 26.15, 19, 55290 etc... Only λ varies "per point".

Note that this is simply a matter of vector addition of 2 vectors.

So, indeed, this vector notation is able to fully describe this line, just like the counterpart

from calculus does (y=ax+b).

Note: the first vector is often called "position vector", and the second (in the direction of the line),

is often called the "direction vector".

A similar vector notation is ofcourse possible for the blue line. See below:

┌ x ┐ └ y ┘ |
= |
┌ 4 ┐ └ 2 ┘ |
+ | μ |
┌ -1┐ └ 2 ┘ |

Task: Try to verify that this notation is correct, using figure 1.

Here, ofcourse the number μ has a similar purpose as λ.

that we now have one additional spatial dimension.

Take a look at figure 2 below:

Figure 2: A line in R

An example of a line in 3Dim space, in vector representation, is:

┌ x ┐ │ y │ └ z ┘ |
= |
┌ 2 ┐ │ 4 │ └ 5 ┘ |
+ | λ |
┌ 1 ┐ │ 1 │ └ 2 ┘ |

the point of intersection of the two lines:

y = 3/2 x + 3 (the red one)

y = -2x + 10 (the blue one)

Obviously, where the lines intersect, the "y" (and "x") of both lines are the same.

So, we must solve:

3/2 x + 3 = -2x + 10 =>

3x + 6 = -4x + 20 =>

7x = 14 =>

x = 2.

We can find the corresponding "y" by using either one of the equations, and substitute "x=2".

Then we will find "y=6". So the point where both lines intersect is (2,6).

We have seen this theory before in "note 2: Linear Equations".

Now, we will do the same, but this time with the vector representations:

┌ x ┐ └ y ┘ |
= |
┌ -4 ┐ └ -3 ┘ |
+ | λ |
┌ 2 ┐ └ 3 ┘ |

┌ x ┐ └ y ┘ |
= |
┌ 4 ┐ └ 2 ┘ |
+ | μ |
┌ -1┐ └ 2 ┘ |

At the intersection, the "x" and "y" at that point must be the same for both lines.

Thus we have:

-4 + 2λ = 4 - μ (equation 1)

-3 + 3λ = 2 + 2μ (equation 2)

This is a set of two equations with two variables. We can solve it in the following way:

I want to isolate either λ or μ. I can do it like so: Multiply equation 1 by "2", and then add both equations:

-8 + 4λ = 8 - 2μ

-3 + 3λ = 2 + 2μ

---------------------------

-11 + 7λ = 10

-11 + 7λ = 10 =>

λ = 3.

Now, I may use the vector equation:

┌ x ┐ └ y ┘ |
= |
┌ -4 ┐ └ -3 ┘ |
+ | 3 * |
┌ 2 ┐ └ 3 ┘ |
= |
┌ -4 ┐ └ -3 ┘ |
+ |
┌ 6 ┐ └ 9 ┘ |
= |
┌ 2 ┐ └ 6 ┘ |

This is indeed the vector (from Origin) pointing to (2,6). So, we have found the point of intersection

of both lines. Note that the traditional representation (2,6), and the "vector", are equivalent in this context.

if the inner product

The inner product is just a number (a scalar), and in case the vectors are not perpendicular, then

the inner product produces a number NOT equal to "0".

For vectors in R

A = |
┌ a ┐ └ b ┘ |

B = |
┌ c ┐ └ d ┘ |

Then A.B = a*c + b*d

Let A and B the vectors:

A = |
┌ 1 ┐ └ 2 ┘ |

B = |
┌ 3 ┐ └ -2┘ |

Then A.B = 1*3 + 2*-2 = -1

But if two vectors are perpendicular, the inner product is "0".

Thus: suppose we have the vector "A":

A = |
┌ 1 ┐ └ 2 ┘ |

Now we want to find the vector A

we may quickly "guess" that the perpendicular vector is:

A^{P} = |
┌ -2┐ └ 1 ┘ |

Let's check that. We calculate the inner product as: 1*-2 + 2*1 = -2+2 = 0.

So, we have indeed found the vector which is perpendicular.

Indeed. If we have:

A = |
┌ a ┐ └ b ┘ |

Then the following vector is perpendicular.

B = |
┌ -b┐ └ a ┘ |

Also, we see that in this case a*-b + b*a = 0.

One application is the following. If you have the vector representation of a certain line,

then you ofcourse have the "direction vector" in that representation too.

Now, you can immediately find the "direction vector" which is perpendicular. Just apply the theory above.

So, now you are almost done in finding the equation of the line which is perpendicular to the first line.

All you need now, is just finding (or locating) some point (which can be any point), which

lies on that second line. If you have that too, you can easily write down the vector representation

of that perpendicular line.

The next note as about quadratic functions and higher order polynomials.