Week of March 9, 2020 College Readiness Math

Systems of Linear Equations

A Linear Equation is an equation for a line.

A linear equation is not always in the form y = 3.5 − 0.5x,

It can also be like y = 0.5(7 − x)

Or like y + 0.5x = 3.5

Or like y + 0.5x − 3.5 = 0 and more.

(Note: those are all the same linear equation!)

A System of Linear Equations is when we have two or more linear equations working together.

Example: Here are two linear equations:

2x	+	y	=	5
−x	+	y	=	2

Together they are a system of linear equations.
Can you discover the values of x and y yourself? (Just have a go, play with them a bit.)

Let's try to build and solve a real world example:

Example: You versus Horse

It's a race!
You can run 0.2 km every minute.
The Horse can run 0.5 km every minute. But it takes 6 minutes to saddle the horse.

How far can you get before the horse catches you?

We can make two equations (d=distance in km, t=time in minutes)

• You run at 0.2km every minute, so d = 0.2t
• The horse runs at 0.5 km per minute, but we take 6 off its time: d = 0.5(t−6)

So we have a system of equations (that are linear):

• d = 0.2t
• d = 0.5(t−6)

We can solve it on a graph:

Do you see how the horse starts at 6 minutes, but then runs faster?
It seems you get caught after 10 minutes ... you only got 2 km away.
Run faster next time.

So now you know what a System of Linear Equations is.

Let us continue to find out more about them ....

Solving

There can be many ways to solve linear equations!

Let us see another example:

Example: Solve these two equations:

• x + y = 6
• −3x + y = 2

The two equations are shown on this graph:

Our task is to find where the two lines cross.

Well, we can see where they cross, so it is already solved graphically.

But now let's solve it using Algebra!

Hmmm ... how to solve this? There can be many ways! In this case both equations have "y" so let's try subtracting the whole second equation from the first:

x + y − (−3x + y) = 6 − 2

Now let us simplify it:

x + y + 3x − y = 6 − 2

4x = 4

x = 1

So now we know the lines cross at x=1.
And we can find the matching value of y using either of the two original equations (because we know they have the same value at x=1). Let's use the first one (you can try the second one yourself):

x + y = 6

1 + y = 6

y = 5

And the solution is:

x = 1 and y = 5

And the graph shows us we are right!

Linear Equations

Only simple variables are allowed in linear equations. No x², y³, √x, etc:

Linear vs non-linear

Dimensions

A Linear Equation can be in 2 dimensions ... (such as x and y)
... or in 3 dimensions ... (it makes a plane)
... or 4 dimensions ...
... or more!

Common Variables

For the equations to "work together" they share one or more variables:

A System of Equations has two or more equations in one or more variables

Many Variables

So a System of Equations could have many equations and many variables.

Example: 3 equations in 3 variables

2x	+	y	−	2z	=	3
x	−	y	−	z	=	0
x	+	y	+	3z	=	12

There can be any combination:

• 2 equations in 3 variables,
• 6 equations in 4 variables,
• 9,000 equations in 567 variables,
• etc.

Solutions

When the number of equations is the same as the number of variables there is likely to be a solution. Not guaranteed, but likely.

In fact there are only three possible cases:

• No solution
• One solution
• Infinitely many solutions

When there is no solution the equations are called "inconsistent".

One or infinitely many solutions are called "consistent"

Here is a diagram for 2 equations in 2 variables:

Independent

"Independent" means that each equation gives new information.
Otherwise they are "Dependent".

Also called "Linear Independence" and "Linear Dependence"

Example:

• x + y = 3
• 2x + 2y = 6

Those equations are "Dependent", because they are really the same equation, just multiplied by 2.
So the second equation gave no new information.

Where the Equations are True

The trick is to find where all equations are true at the same time.

True? What does that mean?

Example: You versus Horse

The "you" line is true all along its length (but nowhere else).
Anywhere on that line d is equal to 0.2t

• at t=5 and d=1, the equation is true (Is d = 0.2t? Yes, as 1 = 0.2×5 is true)
• at t=5 and d=3, the equation is not true (Is d = 0.2t? No, as 3 = 0.2×5 is not true)

Likewise the "horse" line is also true all along its length (but nowhere else).
But only at the point where they cross (at t=10, d=2) are they both true.

So they have to be true simultaneously ...

... that is why some people call them "Simultaneous Linear Equations"

Solve Using Algebra

It is common to use Algebra to solve them.

Here is the "Horse" example solved using Algebra:

Example: You versus Horse

The system of equations is:

• d = 0.2t
• d = 0.5(t−6)

In this case it seems easiest to set them equal to each other:

d = 0.2t = 0.5(t−6)

Start with:0.2t = 0.5(t − 6)

Expand 0.5(t−6):0.2t = 0.5t − 3

Subtract 0.5t from both sides:−0.3t = −3

Divide both sides by −0.3:t = −3/−0.3 = 10 minutes

Now we know when you get caught!

Knowing t we can calculate d:d = 0.2t = 0.2×10 = 2 km

And our solution is:

t = 10 minutes and d = 2 km

Algebra vs Graphs

Why use Algebra when graphs are so easy? Because:

More than 2 variables can't be solved by a simple graph.

So Algebra comes to the rescue with two popular methods:

• Solving By Substitution
• Solving By Elimination

We will see each one, with examples in 2 variables, and in 3 variables. Here goes ...

Solving By Substitution

These are the steps:

• Write one of the equations so it is in the style "variable = ..."
• Replace (i.e. substitute) that variable in the other equation(s).
• Solve the other equation(s)
• (Repeat as necessary)

Here is an example with 2 equations in 2 variables:

Example:

• 3x + 2y = 19
• x + y = 8

We can start with any equation and any variable.
Let's use the second equation and the variable "y" (it looks the simplest equation).

Write one of the equations so it is in the style "variable = ...":

We can subtract x from both sides of x + y = 8 to get y = 8 − x. Now our equations look like this:

• 3x + 2y = 19
• y = 8 − x

Now replace "y" with "8 − x" in the other equation:

• 3x + 2(8 − x) = 19
• y = 8 − x

Solve using the usual algebra methods:

Expand 2(8−x):

• 3x + 16 − 2x = 19
• y = 8 − x

Then 3x−2x = x:

• x + 16 = 19
• y = 8 − x

And lastly 19−16=3

• x = 3
• y = 8 − x

Now we know what x is, we can put it in the y = 8 − x equation:

• x = 3
• y = 8 − 3 = 5

And the answer is:

x = 3
y = 5

Note: because there is a solution the equations are "consistent"

Check: why don't you check to see if x = 3 and y = 5 works in both equations?

Solving By Substitution: 3 equations in 3 variables

OK! Let's move to a longer example: 3 equations in 3 variables.

This is not hard to do... it just takes a long time!

Example:

• x + z = 6
• z − 3y = 7
• 2x + y + 3z = 15

We should line up the variables neatly, or we may lose track of what we are doing:

x			+	z	=	6
	−	3y	+	z	=	7
2x	+	y	+	3z	=	15

WeI can start with any equation and any variable. Let's use the first equation and the variable "x".

Write one of the equations so it is in the style "variable = ...":

x					=	6 − z
	−	3y	+	z	=	7
2x	+	y	+	3z	=	15

Now replace "x" with "6 − z" in the other equations:

(Luckily there is only one other equation with x in it)

	x					=	6 − z
		−	3y	+	z	=	7
2(6−z)		+	y	+	3z	=	15

Solve using the usual algebra methods:

2(6−z) + y + 3z = 15 simplifies to y + z = 3:

x					=	6 − z
	−	3y	+	z	=	7
		y	+	z	=	3

Good. We have made some progress, but not there yet.

Now repeat the process, but just for the last 2 equations.

Write one of the equations so it is in the style "variable = ...":

Let's choose the last equation and the variable z:

x					=	6 − z
	−	3y	+	z	=	7
				z	=	3 − y

Now replace "z" with "3 − y" in the other equation:

x					=	6 − z
	−	3y	+	3 − y	=	7
				z	=	3 − y

Solve using the usual algebra methods:

−3y + (3−y) = 7 simplifies to −4y = 4, or in other words y = −1

x					=	6 − z
		y			=	−1
				z	=	3 − y

Almost Done!

Knowing that y = −1 we can calculate that z = 3−y = 4:

x					=	6 − z
		y			=	−1
				z	=	4

And knowing that z = 4 we can calculate that x = 6−z = 2:

x					=	2
		y			=	−1
				z	=	4

And the answer is:

x = 2
y = −1
z = 4

Check: please check this yourself.

We can use this method for 4 or more equations and variables... just do the same steps again and again until it is solved.

Conclusion: Substitution works nicely, but does take a long time to do.

Solving By Elimination

Elimination can be faster ... but needs to be kept neat.

"Eliminate" means to remove: this method works by removing variables until there is just one left.

The idea is that we can safely:

• multiply an equation by a constant (except zero),
• add (or subtract) an equation on to another equation

Like in these examples:

WHY can we add equations to each other?

Imagine two really simple equations:

x − 5 = 3
5 = 5

We can add the "5 = 5" to "x − 5 = 3":

x − 5 + 5 = 3 + 5
x = 8

Try that yourself but use 5 = 3+2 as the 2nd equation
It will still work just fine, because both sides are equal (that is what the = is for!)

We can also swap equations around, so the 1st could become the 2nd, etc, if that helps.

OK, time for a full example. Let's use the 2 equations in 2 variables example from before:

Example:

• 3x + 2y = 19
• x + y = 8

Very important to keep things neat:

3x	+	2y	=	19
x	+	y	=	8

Now ... our aim is to eliminate a variable from an equation.

First we see there is a "2y" and a "y", so let's work on that.
Multiply the second equation by 2:

3x	+	2y	=	19
2x	+	2y	=	16

Subtract the second equation from the first equation:

x			=	3
2x	+	2y	=	16

Yay! Now we know what x is!

Next we see the 2nd equation has "2x", so let's halve it, and then subtract "x":
Multiply the second equation by ½ (i.e. divide by 2):

x			=	3
x	+	y	=	8

Subtract the first equation from the second equation:

x			=	3
		y	=	5

Done!
And the answer is:

x = 3 and y = 5

And here is the graph:

The blue line is where 3x + 2y = 19 is true

The red line is where x + y = 8 is true

At x=3, y=5 (where the lines cross) they are both true. That is the answer.

Here is another example:

Example:

• 2x − y = 4
• 6x − 3y = 3

Lay it out neatly:

2x	−	y	=	4
6x	−	3y	=	3

Multiply the first equation by 3:

6x	−	3y	=	12
6x	−	3y	=	3

Subtract the second equation from the first equation:

0	−	0	=	9
6x	−	3y	=	3

0 − 0 = 9 ???

What is going on here?

Quite simply, there is no solution.

They are actually parallel lines:

And lastly:

Example:

• 2x − y = 4
• 6x − 3y = 12

Neatly:

2x	−	y	=	4
6x	−	3y	=	12

Multiply the first equation by 3:

6x	−	3y	=	12
6x	−	3y	=	12

Subtract the second equation from the first equation:

0	−	0	=	0
6x	−	3y	=	3

0 − 0 = 0

Well, that is actually TRUE! Zero does equal zero ...

... that is because they are really the same equation ...

... so there are an Infinite Number of Solutions

They are the same line:

And so now we have seen an example of each of the three possible cases:

• No solution
• One solution
• Infinitely many solutions

Solving By Elimination: 3 equations in 3 variables

Before we start on the next example, let's look at an improved way to do things.

Follow this method and we are less likely to make a mistake.

First of all, eliminate the variables in order:

• Eliminate xs first (from equation 2 and 3, in order)
• then eliminate y (from equation 3)

So this is how we eliminate them:

We then have this "triangle shape":

Now start at the bottom and work back up (called "Back-Substitution")
(put in z to find y, then z and y to find x):

And we are solved:

ALSO, we will find it is easier to do some of the calculations in our head, or on scratch paper, rather than always working within the set of equations:

Example:

• x + y + z = 6
• 2y + 5z = −4
• 2x + 5y − z = 27

Written neatly:

x	+	y	+	z	=	6
		2y	+	5z	=	−4
2x	+	5y	−	z	=	27

First, eliminate x from 2nd and 3rd equation.

There is no x in the 2nd equation ... move on to the 3rd equation:
Subtract 2 times the 1st equation from the 3rd equation (just do this in your head or on scratch paper):

And we get:

x	+	y	+	z	=	6
		2y	+	5z	=	−4
		3y	−	3z	=	15

Next, eliminate y from 3rd equation.

We could subtract 1½ times the 2nd equation from the 3rd equation (because 1½ times 2 is 3) ...
... but we can avoid fractions if we:

• multiply the 3rd equation by 2 and
• multiply the 2nd equation by 3

and then do the subtraction ... like this:

And we end up with:

x	+	y	+	z	=	6
		2y	+	5z	=	−4
				z	=	−2

We now have that "triangle shape"!

Now go back up again "back-substituting":

We know z, so 2y+5z=−4 becomes 2y−10=−4, then 2y=6, so y=3:

x	+	y	+	z	=	6
		y			=	3
				z	=	−2

Then x+y+z=6 becomes x+3−2=6, so x=6−3+2=5

x					=	5
		y			=	3
				z	=	−2

And the answer is:

x = 5
y = 3
z = −2

Check: please check for yourself.

General Advice

Once you get used to the Elimination Method it becomes easier than Substitution, because you just follow the steps and the answers appear.

But sometimes Substitution can give a quicker result.

• Substitution is often easier for small cases (like 2 equations, or sometimes 3 equations)
• Elimination is easier for larger cases

Source:https://www.mathsisfun.com/algebra/systems-linear-equations.html

Week of March 9, 2020 Algebra 1

Greatest Common Factor

The highest number that divides exactly into two or more numbers.
It is the "greatest" thing for simplifying fractions!

Let's start with an Example ... 

Greatest Common Factor of 12 and 16

1. Find all the Factors of each number,
2. Circle the Common factors,
3. Choose the Greatest of those

So ... what is a "Factor" ?

Factors are numbers we can multiply together to get another number:

A number can have many factors:

Factors of 12 are 1, 2, 3, 4, 6 and 12 ...

... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.

(Read how to find All the Factors of a Number. In our case we don't need the negative ones.)

What is a "Common Factor" ?

Say we have worked out the factors of two numbers:

Example: Factors of 12 and 30

Factors of 12 are 1, 2, 3, 4, 6 and 12

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Then the common factors are those that are found in both lists:

• Notice that 1, 2, 3 and 6 appear in both lists?
• So, the common factors of 12 and 30 are: 1, 2, 3 and 6

It is a common factor when it is a factor of two (or more) numbers.

Here is another example with three numbers:

Example: The common factors of 15, 30 and 105

Factors of 15 are 1, 3, 5, and 15

Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105

The factors that are common to all three numbers are 1, 3, 5 and 15
In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15

What is the "Greatest Common Factor" ?

It is simply the largest of the common factors.

In our previous example, the largest of the common factors is 15, so the Greatest Common Factor of 15, 30 and 105 is 15

The "Greatest Common Factor" is the largest of the common factors (of two or more numbers)

Why is this Useful?

One of the most useful things is when we want to simplify a fraction:

Example: How can we simplify 1230 ?

Earlier we found that the Common Factors of 12 and 30 are 1, 2, 3 and 6, and so the Greatest Common Factor is 6.

So the largest number we can divide both 12 and 30 exactly by is 6, like this:

	÷ 6
1230	=	25
	÷ 6

The Greatest Common Factor of 12 and 30 is 6.
And so 1230 can be simplified to 25

Finding the Greatest Common Factor

Here are three ways:

1. We can:

• find all factors of both numbers (use the All Factors Calculator),
• then find the ones that are common to both, and
• then choose the greatest.

Example:

Two Numbers	Factors	Common Factors	Greatest Common Factor	Example Simplified Fraction
9 and 12	9: 1,3,9 12: 1,2,3,4,6,12	1,3	3	912 = 34

And another example:

Two Numbers	Factors	Common Factors	Greatest Common Factor	Example Simplified Fraction
6 and 18	6: 1,2,3,6 18: 1,2,3,6,9,18	1,2,3,6	6	618 = 13

2. Or we can find the prime factors and combine the common ones together:

Two Numbers	Thinking ...	Greatest Common Factor	Example Simplified Fraction
24 and 108	2 × 2 × 2 × 3 = 24, and 2 × 2 × 3 × 3 × 3 = 108	2 × 2 × 3 = 12	24108 = 29

3. Or sometimes we can just play around with the factors until we discover it:

Two Numbers	Thinking ...	Greatest Common Factor	Example Simplified Fraction
9 and 12	3 × 3 = 9 and 3 × 4 = 12	3	912 = 34

But in that case we must check that we have found the greatest common factor.

Greatest Common Factor Calculator

OK, there is also a really easy method: we can use the Greatest Common Factor Calculator to find it automatically.

Source:https://www.mathsisfun.com/greatest-common-factor.html

Week of March 2, 2020 Algebra 1

Solving Inequalities

Sometimes we need to solve Inequalities like these:

Symbol	Words	Example
>	greater than	x + 3 > 2
<	less than	7x < 28
≥	greater than or equal to	5 ≥ x − 1
≤	less than or equal to	2y + 1 ≤ 7

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

Something like:		x < 5
or:		y ≥ 11

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10

Solved!

How to Solve

Solving inequalities is very like solving equations ... we do most of the same things ...

... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things can change the direction!

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

Safe Things To Do

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ("<" becomes ">" for example):

Multiply (or divide) both sides by a negative number
Swapping left and right hand sides

Example: 2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality:

12 > 2y+7

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Solve: x + 3 < 7

If we subtract 3 from both sides, we get:

x + 3 − 3 < 7 − 3    

x < 4

And that is our solution: x < 4
In other words, x can be any value less than 4.

What did we do?

We went from this: To this:				x+3 < 7 x < 4

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5
If we subtract 5 from both sides, we get:

12 − 5 < x + 5 − 5    

7 < x

That is a solution!
But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:

Solve: 3y < 15

If we divide both sides by 3 we get:

3y/3 < 15/3

y < 5

And that is our solution: y < 5

Negative Values

When we multiply or divide by a negative number we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from −3 to −7 is a decrease.

−7 < −3	7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Solve: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality!

−2y < −8

−2y/−2 > −8/−2

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Solve: bx < 3b

It seems easy just to divide both sides by b, which gives us:

x < 3

... but wait ... if b is negative we need to reverse the inequality like this:

x > 3

But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:

• if b is 1, then the answer is x < 3
• but if b is −1, then we are solving −x < −3, and the answer is x > 3

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.

So:

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

Solve: x−32 < −5

First, let us clear out the "/2" by multiplying both sides by 2.
Because we are multiplying by a positive number, the inequalities will not change.

x−32 ×2 < −5 ×2  

x−3 < −10

Now add 3 to both sides:

x−3 + 3 < −10 + 3    

x < −7

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Solve:

−2 < 6−2x3 < 4

First, let us clear out the "/3" by multiplying each part by 3.
Because we are multiplying by a positive number, the inequalities will not change:

−6 < 6−2x < 12

Now subtract 6 from each part:

−12 < −2x < 6

Now multiply each part by −(1/2).
Because we are multiplying by a negative number, the inequalities change direction.

6 > x >−3

And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

−3 < x < 6

Summary

• Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
• But these things will change direction of the inequality:
  • Multiplying or dividing both sides by a negative number
  • Swapping left and right hand sides
• Don't multiply or divide by a variable (unless you know it is always positive or always negative)

 Source:https://www.mathsisfun.com/algebra/inequality-solving.html

Week of March 2, 2020 College Readiness Math

Point-Slope Equation of a Line

The "point-slope" form of the equation of a straight line is:

y − y₁ = m(x − x₁)

The equation is useful when we know:

• one point on the line: (x₁,y₁)
• and the slope of the line: m,

and want to find other points on the line.

Have a play with it first (move the point, try different slopes):

y − 6 = 0.5(x − 8)

Slope:

0.5

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Now let's discover more.

What does it stand for?

(x₁, y₁) is a known point

m is the slope of the line

(x, y) is any other point on the line

Making sense of it

It is based on the slope:

Slope m  =  change in ychange in x  =  y − y₁x − x₁

Starting with the slope: we rearrange it like this: to get this:

So, it is just the slope formula in a different way!

Now let us see how to use it.

Example 1:

slope "m"  =  31  =  3

y − y₁ = m(x − x₁)

We know m, and also know that (x₁, y₁) = (3,2), and so we have:

y − 2 = 3(x − 3)

That is a perfectly good answer, but we can simplify it a little:

y − 2 = 3x − 9

y = 3x − 9 + 2

y = 3x − 7

Example 2:

m = −31 = −3

y − y₁ = m(x − x₁)

We can pick any point for (x₁, y₁), so let's choose (0,0), and we have:

y − 0 = −3(x − 0)

Which can be simplified to:

y = −3x

Example 3: Vertical Line

What is the equation for a vertical line?
The slope is undefined!

In fact, this is a special case, and we use a different equation, like this:

x = 1.5

Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5

What About y = mx + b ?

You may already be familiar with the "y=mx+b" form (called the slope-intercept form of the equation of a line).

It is the same equation, in a different form!

The "b" value (called the y-intercept) is where the line crosses the y-axis.

So point (x₁, y₁) is actually at (0, b)

and the equation becomes:

Start withy − y₁ = m(x − x₁)

(x₁, y₁) is actually (0, b):y − b = m(x − 0)

Which is:y − b = mx

Put b on other side:y = mx + b

source: https://www.mathsisfun.com/algebra/line-equation-point-slope.html