Solve Systems of Linear Equations: Intro

This lesson is the first of a four part series about systems of linear equations. Here, I define what a system of equations is, and the three types of solutions. Future lessons will cover graphing, elimination, and substitution.

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transcript:

Welcome to Your Tutor Online video lessons. This is the first of a four part series on solving systems of linear equations. This first lesson will serve as an introduction.

A system means a group, or more than one. And the word linear here, refers to a line, and also means we will be dealing with two equations which each have two variables in them. Normally they will be x and y. We need to work with both equations at the same time to get a solution. Here's an example of what a system of equations problem looks like. 3x minus 4y is equal to 2 and 7x plus 3y is equal to 4. Now, using the ways I'll show you in future lessons you will be able to work with these two equations and get a solution for x and y.

There are three types of solutions to this kind of problem. You can have one solution, no solutions, or infinite solutions.

You have one solution when the two lines intersect. The point where the lines intersect is your solution. You can give the answer as an ordered pair, in this case "(4,2)" or you can split up the coordinates and say that x is equal to its coordinate which is 4 and y is equal to its coordinate which is 2.

A system has no solutions if the lines are parallel or never intersect. This can be seen clearly with graphing. Or, prior to graphing if the two lines have the same slope but not the same y intercept then they are parallel and have no solution. You can simply write "no solution" or this symbol, a zero with a slash through it, which means the same thing.

There are infinite solutions if the two lines you are dealing with are actually the same line. Here is an example. I have a blue line and a green line right on top of each other. This can happen if the first equation you are given is a multiple of the second or vice versa. We'll take a closer look at that in upcoming lessons. These two are the same line, so they'll intersect at every single point. So, we say there's an infinite number of solutions.

There are three ways to solve systems of linear equations: By graphing, elimination, and substitution. Each of these will be the focus of the next three lessons. Be sure to visit www.YourTutorOnline.com to see them when they become available. And, as always, if you have any questions leave a comment underneath any of the lessons. Thanks for watching, class dismissed.

Video Series on Systems of Equations

I will be putting up 4 new video lessons this week! They will all look at how to solve systems of linear equations.

• Lesson 1, on Tuesday, will serve as an introduction
• Lesson 2, on Wednesday, will look at solving by graphing
• Lesson 3, on Thursday, will cover elimination
• Lesson 4, on Friday, will wrap it up with substitution
  

Lesson 1 will be posted tomorrow morning, bright and early.

Simplify Radicals

This lessons looks at how to simplify radical expressions which are not perfect squares or cubes. I include an example of square root, cube root, and 4th root. I show a technique to simplify n-root expressions. Also I look at how to simplify variables in radical expressions.

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transcript:

Welcome to Your Tutor Video lessons. Today we'll look at how to simplify radicals.

A radical is anything with a radical symbol which looks like this. It can be square root, cube root, fourth root, or any numbered root. If there is no number in the symbol then its a square root, otherwise there will be a tiny number here to show you what root it is. In this example it is cube root. These expressions need to be simplified like anything else in math.

Lets look at this example: radical 54 or square root of 54. The easiest way I have found to simplify radicals is to do a factor tree for the terms inside the radical. Now this only works if everything inside the radical is multiplied together or you only have one term.

First draw two lines out from the number or term inside the radical. And on this line we are going to write any two numbers that multiply together to give us our originial. I'm going to choose 9 and 6. We are going to repeat the process until we have nothing left except for prime numbers. 3 times 3 is equal to 9 and 3 times 2 is equal to 6. Now we are left with only prime numbers here at the bottom, 2s and 3s.

Now we want to look and see which root we are dealing with. Here it is the square root or 2nd root because there is no number in the symbol itself. Since its the second root, we are looking for groups of two of the same numbers. Here I see two-3s. When you get a group, you can write the group number outside of the radical, in this case its 3. Then we'll go ahead and write our radical and write everything thats left over inside, multiplied together: 3 times 2. Go ahead and multiply those together: 3 radical 6, and thats our simplified expression.

You can look for shortcuts as you do your factor tree. As you get more practice with this type of problem, you will know that 9 is a perfect square and so it's square root can automatically come out of the radical. For cube roots you are looking for perfect cubes and so on according to whatever radical you happen to be dealing with. Now its going to get harder as you get into higher radicals and you probably won't have high powers memorized, but thats okay you can just do your factor tree when in doubt.

Lets look at how this works for a cube root. We'll use the example cube root 135. We are still going to go the factor tree just like we did in the previous example. 5 times 27 is equal to 135. 5 is a prime number so we will just leave that alone. 27 is 3 times 9 and 9 is 3 times 3. Notice we could of had a shortcut there if you knew that 27 was a perfect cube.

This time we are going to look for groups of three because we are dealing with the cube root. Here you can see three-3s. So that means a 3 is allowed to come out of the radical. So I write 3, the radical symbol with its cube root and whatever is left over that didn't find a group of three, which is 5. So this radical simplified is 3 cube roots of 5.

Lets look at how to do radicals with variables in them. For a coefficient we are still going to do a factor tree. 48 is 12 times 4. 12 is 3 times 4. 3 is a prime number. 4 is 2 times 2. And this 4 is 2 times 2. This time we are going to be looking for groups of four because we are dealing with the fourth root. So here, I found four-2s. So the two is going to come out and a 3 will stay inside.

Now lets look at our variables. This part is a little tricky. We are going to look at each of our exponents for our variables. We are going to take our root number and divide it into each of the exponents. The number of times that the root goes into the exponent, will be our new exponent on the outside of the radical and the remainder will be the new exponent that stays behind inside the radical.

Now we can get started on our answer. We know a 2 comes out so I'm going to write that. I'll leave some space, draw my radical symbol with is the fourth root. I know a 3 is stuck inside. Now I can start dealing with the variables.

4 goes into 6 one time so x to the first power is on the outside. 4 goes into 6 once and there are 2 left over, so its x squared left on the inside. 4 does not go into 2 any so a y does not come out of the radical. 4 goes into 2 zero times with 2 left over so we have a y squared left on the inside. 4 goes into 8 two times so we have z squared on the outside. 4 goes into 8 twice with no remaineder so there is no z left on the inside. And this is our final answer: 2 x z squared times the fourth root of 3 x squared y squared.

Thanks for watching Your Tutor video lessons, now I know today was a little bit of a tricky topic so if you have any questions head over to the blog and leave a question at www.YourTutorOnline.com And also, if you haven't been there yet be sure to check out the quizzes I'm going to put up after each lesson and see how much you understand. Thanks for watching, class dismissed.

Solve Equations with Absolute Value

This lesson shows how to solve equations involving absolute value.

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transcript:

Welcome to Your Tutor Online video lessons. Today we will learn how to solve equations that have an absolute value in them.

As a review, absolute value refers to how far away a number is from zero. To put it simply, the absolute value makes a number positive no matter what. Three is three units away from zero. Negative four is four units away from zero. So the absolute value of three is three, and the absolute value of negative four is four. Three stays positive, number four becomes positive.

What if there is a variable inside the absolute value sign? Well, in our order of operations, we want to treat the absolute value sign just like we would parentheses. So simplify everything inside them first, and move everything you can to the other side. In this example, 3 times the absolute value of x plus 2 is equal to 9, we can divide both sides by 3 and we are left with the absolute value of x plus 2 is equal to 3. When you get the absolute value all by itself on one side, you are done with the simplification.

Now we need to split this equation into two separate equations. For the first one, simply drop the absolute value signs. We have x plus 2 is equal to 3. For the other equation drop the absolute value signs and this time make the other side of the equation negative. Or if its already negative make it positive, just flip the sign of the other side of the equation.

Solve each equation that you came up with. On the left x is equal to 1 and on the right x is equal to negative 5. So for absolute value equations we will have two solutions, In this case x is equal to 1 or x is negative 5.

I hope you found this lesson useful. If you have any questions leave a comment on the blog at www.YourTutorOnline.com If you have any lessons suggestions send an email to podcast[at]yourtutoronline.com Thanks for watching, class dismissed.

Points and Lines

This lesson covers how to identify, name and label points and lines.

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transcript:

Welcome to Your Tutor Online video lessons. Today we will learn how to identify, name and label points and lines.

A point does not take up any space, but so that we can see it we draw a dot. Points are labeled with capital letters. You call a point by its letter, and is read "point A."

Two points make up a line, and a line extends forever in both directions. We can represent that by drawing arrows on either side. To name a line pick any two points that are on the line. Here we have points A and B. Lines are also labeled with capital letters, so this is called "line AB." It can be any letters on the line at all. If we had another point here, C, it can be "line AC", "line CA", "line CB", all those are fine labels for this line. The symbol for a line is the two capital letters with a line over top, just remember to draw the arrows. We read this "line AB." You can also label a line with a lower case letter. For example we can say this is "line n."

Points can be collinear or non-collinear. Collinear just means that the points are all on the same line. A,B, and C are all collinear points. Points D, E, and F are non-collinear because you cannot draw a line that will connect all the points.

A line segment is part of line, has end points, and does not go on forever. Line segments can be measured. You name a line segment by its end points. For example here we have line segment AB. The symbol for a segment is the capital letters of the two end points with a bar on top. The order here doesn't matter. Segment AB is the same as segment BA.

Because line segments can be measured they can be compared with one another. If two lines segments are the same lenght they are called congruent. The symbol for congruent is an equal sign with a funny hat on top. An up and down dash can also represent congruency. Segment AB is congruent to segment BC. When this happens we know that B is the midpoint of segment AC because it divides that segment into two equal parts.

When we are comparing things, such as line segments, we use the word congruent. When we are comparing measurements we use the word equal. The measurement of segment AB is equal to the measurement of segment BC.

A ray has an endpoint and extends forever in one direction. To name a ray always start with the end point and then pick one other point on the ray. This example can be called ray AH. The symbol for a ray is going to be the capital letters of the end point (first) and then any other point with a tiny ray above it. The ray above it will always point to the right regardless of the way the ray actually points.

I hope you found this lesson useful. If you have any questions leave a comment on the blog at www.YourTutorOnline.com If you have any lessons suggestions send an email to podcast[at]yourtutoronline.com Thanks for watching, class dismissed.

Graph Linear Inequalities

How to graph a linear inequality. When to use a dotted, or solid line, and which side to shade on.

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transcript:

Welcome to Your Tutor Online video lessons. Today will we learn how to graph linear inequalities. A linear inequality is an equation for a line. That means it will have two variables and it has a less than, greater than, less than or equal to, or great than or equal to sign instead of the equals sign. Here we have an example 2x minus 3y is less than 6.

For the most part we are just going to pretend that this less than sign is an equal sign to solve the equation into slope intercept form. The only exception is if we multiple or divide by a negative number. When that happens, flip the sign so that it points in the other direction.

To understand why we need to flip the sign lets look at this very simply example. 3 is less than 5 which is true. Now lets multiply everything by negative 1. In a normal equation this would be fine, as long as you do something to one side that you do to the other you should maintain the equation. But here we'll get negative 3 is less than negative 5 which is not true. Negative 5 is less than negative 3. So to account for that, instead we are going to flip the sign whenever we multiply or divide by a negative. Negative 3 is greater than negative 5, which is true.

Now let's go back to our example and put it into slope intercept form. We are going to subtract 2x from both sides. That leaves us with negative 3y is less than negative 2x plus 6. Now we divide by negative 3 and we are left with y on this side. We need to flip our sign since we are dividing by a negative number. So now it's greater than. Negative 2 divided by negative 3 is positve two-thirds-x and 6 divided by negative 3 is negative 2.

Now that the equation is in slope intercept form, graph it as you normally would. I'm going to start at the y axis, negative 2 for the intercept, is here. Now we are going to follow our slope - up two, over three. We are going to do that one more time: up two over three. Now, before we connect the dots we need to go back and look at the equation. Graphing here is slightly different than a normal line. If the inequality has a less than or greater than sign, we will use the dashed line. If it is less than or eqaul to, or greater than or equal to we will use a solid line. If "equal" is in the name of the symbol then use a normal line. For our example, it's greater than so we use the dashed line. We are going to use a dashed line because the points on this line are not included in the equation.

There's one extra thing we need to do when graphing inequalities. We have to shade on one side of the line. To figure out which one, pick any point that is not on the line. I always use the point zero, zero when I can because it is the easiest to work with. Remember that in ordered pairs the first number is x and the second number is y. We are going to take those two numbers and plug them into our inequality.

Zero is greater than zero times two-thirds x is zero; negative two. Zero is greater than negative two. If that statement is true then we will shade on the side of the line that contains the point we tested for. If its false then we will just shade on the opposite side.

So, zero is greater than negative two is true, so we shade on this side of the dotted line. And thats all there is to graphing inequalities.

I hope you found this lesson useful. If you have any questions leave a comment on the blog at www.YourTutorOnline.com If you have any lessons suggestions send an email to podcast[at]yourtutoronline.com Thanks for watching, class dismissed.

Summer Vacation

I am now officially on summer vacation. But, that doesn't mean that I am not hard at work. I will be developing new lessons which will be released this fall.

If you have a topic or lesson you would like to see please let me know. I want to create lessons that people want and need.

Update: (Lessons already produced for the fall)

• Graphing Inequalities
• Points and Lines
• Solve Equations with Absolute Value
• Simplify Radicals

What would you like to see? Click on the "comments" link below to leave your feedback!