algebra essentials

Algebra Essentials

This is the start of a new series involving Algebra and number sets. It will start out with basic Algebra but get more advanced in time.

Chapter 1 - Introduction

I have been wanting to start this series for a long time. I think advanced Algebra is very interesting and I want to explore it as much as I can. However, I need to start out with the basics for those that would not be familiar with it. I have been thinking how far in the beginning I wanted to start. It eventually occurred to me that I wanted to start at the very beginning for the sake of my son. My son is young and has no interest in math yet. I hope that changes soon. This series is supposed to be something that he can eventually follow and learn from if he develops that interest. I will just expose math to him regularly and hope for the best.

Number Patterns

The basics of numbers is about patterns. Whenever we look at a series of numbers a pattern often develops. As we go about our day to day activities we use numbers and the resulting patterns to help us do a better job. Most of the time we do this unconsciously. However, if we think about it a little more we can use this number behavior to our advantage.

First, let us look at numbers from a very basic standpoint so we are all on the same page. Numbers can be rational, irrational, natural or integers.

5 = natural, integer, and rational

-1.2 = rational

13/7 = rational

sqrt -7 = irrational

-12 = integer and rational

sqrt 16 = natural, integer, and rational because it evaluates to 4.

These definitions will help us later on when dealing with certain mathematical situations. Each type of number I have listed above has certain rules about what can be done with them. It is these rules that dictate how we interact with problems to solve them. From here on I will give some basic examples of these points to cement them in your heads.

Example - Find the average of some amount of numbers.

If you have 72 computers with varying numbers on 3 different pallets, how many computers are on each pallet on average?

This is a nice and easy example to start with. Most of you can instantly recognize that the answer is 24. A few might have to bust out a calculator but you understand how to solve the problem at the very least.

Now lets do another example that many adults don’t know how to do but is simple nonetheless. This one is about calculating percent change in two numbers. I asked a couple people and neither adult knew how to do this off top of their heads. They are both proficient with Excel though. Kind of concerning isn’t it? I think we often rely on our tools too much. In fact, many people can’t do things by hand anymore. That is also the reason why I get asked to help with these things, as these people don’t know why the calculations they do work or don’t work. When something does not give the expected results, they can’t debug their own problem to figure out why.

Example - Calculate the percent change in these two numbers: 38 and 57.

I will explain what we are doing so it is clear. First, we get the difference between the two numbers. 57-38 = 19. We then divide the difference by the lesser number. 19/38 = .5. Multiply the .5 times 100 and that is your percent change. .5 * 100 = 50% change.

If you notice, I am going over some common tasks that everyone should know how to do. I am trying to explain how they work also. Another problem someone might be curious about is how fast the Earth moves. So lets work on this.

Example - Calculate the speed of Earth in its orbit.

Let’s think about what we know. Its orbit is the elliptical path it takes around the Sun. An elliptical path is a very stretched out circle in simple terms. The Earth goes around the Sun because of the Sun’s gravitational pull. We also know that the average radius of our orbital path is 93 million miles. The Earth also takes approximately 1 year to travel this path. We will use this simple geometry to figure out the rest of what we need.

Speed is distance divided by time. Our first task is to find out how far we travel in a year. How many miles is it really? We want to use the formula for a circle here because it’s a fair approximation. The distance around a circle is 2* pi * r.

D = 2 * 3.14 * 93,000,000 which is rounded to 584,000,000 miles. Hours in a year = 365 * 24 = 8760.

We have our problem data now so just divide. 584,000,000 / 8760 = 67,000 miles per hour. I rounded that up. That is how fast we are traveling around the Sun.

Another good ability to have is to be able to calculate the volume of common objects. This is handy in cooking, 3d printing, and estimating the mass of stellar objects 50 billion light years away. Fun right! So what would the volume be of a simple soda can?

Example - Find the volume of the infamous soda can!

This soft drink is a cylinder. That is its shape.The formula for the volume of a cylinder is v = pi * r^2 * h. V = volume, pi = 3.14, r = radius, and h = height. If r = 1.4 inches and and h = 5 inches, use these figures to find the volume.

V = (3.14) * (1.4)^2 * (5).

V = 30.78 cubic inches.

Another good tool to have in your mental repertoire is being able to measure the thickness of objects through basic calculations.It’s actually a pretty quick calculation once you understand it. Lets try it out.

Example - What is the thickness of an object that is 15 cm by 35 cm and weighs 5.4 grams?

Thickness = volume / area. 1 cubic cm of this material weighs 2.7 grams. We start by finding the volume. Volume in this case will be length * width * thickness.

We know the material weighs a total of 5.4 grams and also that 1 cubic cm is equal to 2.7 grams. So that means we have 2 cubic cm of material, which is our volume.

The area of this material is easy since it seems to be rectangular. 15 cm * 35 cm = 525 cm ^2.

Thickness = volume / area. Thickness = 2 cm ^3 / 525 cm ^2. Thickness = .0038 cm. That is our answer. Its pretty thin isn’t it? This process does show you how useful it is to be able to do these calculations. You have to have the correct information up front of course.

 

I am going to stop here for this entry. In this topic I talked about the types of numbers we often see. I hopefully explained why it is important to do calculations by hand at first, until they are understood. We did several examples together and I hope they were able to be understood by my explanations. This will be an ongoing series and I think it will be a lot of fun to learn and teach others. Percy, if you read this far, this one was for you my son!

 

 

In this article I discuss the history and uses of functions. Algebraic functions are some of the most important concepts in math. Since this is a series on Algebra, I use specific examples that are related.

Chapter 2 - Introduction

Functions are a type of relation that tie numbers together. They have to have a mutual meaning, but when they do, the data given can be very meaningful. Beginning functions often use tables for a visual aid. Tables in this fashion show the direct relationship between a set of numbers.

Notation and Pairs

When talking about functions, the notation y = f(x) is traditionally used. This means that when (x) is used as an input into (f), then the result is y. That is a fundamental relationship and must be understood. It is important to realize that a function returns a set of ordered pairs, like (x,y). A relation that has one output for each input is, by definition, a function. (x) is the independent variable and (y) is the dependent variable.

A set of inputs (x) is called the domain of the function. The set of related outputs (y) is the range of the function. A letter (f) is usually the name of a function. A function can be represented in words, a table, or a diagram. They all mean the same thing. Each method will have its limitations.

Functions can be represented verbally, graphically, symbolically, or numerically. There is no best way, it depends on the situation and the data you want to show.

Every function is a relation but not every relation is a function. This is important to remember. The domain is the set of all read numbers for which it is defined.

Constant Functions

I want to add some detail to our definition of functions. They can be considered in another light, that is, they can be linear or nonlinear. An example of a linear function is called a constant function. It should not be too hard to surmise what this is. A constant function is one who’s output remains constant. These are actually very common.

Linear Functions

This is a type of function where the output has the same rate of increasing or decreasing. It is represented by f(x) = ax + b. If a=0 at any point then we will just have a constant function. So in a linear function, each time x increases, f(x) changes by the amount of a.

Slope

The graph of a linear function is a straight line. Slope is the number that indicates the inclination of this line. If you have two points, slope = m = \frac{\delta y}{\delta x}. If this slope is positive, it rises from left to the right. However, if its negative, the line sinks from left to right.

Example 1

Find the slope of the line passing through the points (-2,3) and (1, -2).

You calculate it like this:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2-3}{1-(-2)} = \frac{-5}{3} $$

Slope like this is also known as rate of change of a function. It is also a constant function.

When using a graph to figure out estimates, the units will be easy and given for you. It will be the units for the y-axis over the x-axis. Take advantage of this.

Nonlinear Functions

Nonlinear functions are easy to explain. They are functions with variable output. So the output does not increase or decrease with any regular interval. If you were looking at a graph, the line would be curved. This is what a curved line means.

If you are looking at a table of data and wonder if it is linear or nonlinear, just examine the output. That part should be labeled clearly. The output will have the same rate of change if it is linear, otherwise it is nonlinear.

Average Rate of Change

Since the graphs of nonlinear functions are not straight, there is no single slope. Instead, there are many different slopes in a particular curve. However, we can take an average of the slopes to get a general idea. This is an important calculus topic too. If we have two points on a curve and draw a straight line connecting them, we call this a secant line. So the average rate of change is telling us, on average, how fast a quantity is changing. the average rate of change is just the slope of that particular line between two points.

When a function is constant, its average rate of change, or slope, is 0. In a linear function such as:

$$ f(x) = ax + b $$

its average rate of change is equal to a, the slope of its graph. A nonlinear function has a variable rate of change.

Difference Quotient

The difference quotient is encountered a lot in calculus. It looks like this:

$$ \frac{f(x + h) - f(x)}{h} $$

Example 2

Calculating A Difference Quotient

If $$ f(x) = x^2 - 2x $$ Find f(x+h)

Our expression is $$ x^2 - 2x $$ . We substitute (x+h) for every (x) in our expression.

$$ f(x + h) = (x + h)^2 - 2(x + h) $$

Now multiple the expression out.

$$ f(x + h) = x^2 + 2xh + h^2 -2x -2h $$

Ok that gives us $$ f(x + h) $$ . Now we input that into our difference quotient.

$$ \frac{ f(x + h) - f(x)}{h } = \frac{ x^2 + 2xh + h^2 -2x -2h- (x^2 - 2x) } {h} $$

Lets rewrite that ending part so everything has the correct sign.

$$ \frac{ f(x + h) - f(x) } {h} = \frac{ x^2 + 2xh + h^2 -2x -2h -x^2 + 2x } {h} $$

It should be immediately obvious that a few things will cancel out. Lets do that.

$$ \frac{ 2xh + h^2 -2h }{h} $$

$$ \frac{ h(2x + h -2) } {h} $$

$$ 2x + h -2 $$

This is our answer. These problems are usually designed so they cancel out certain values to make it work. So when working these, if the problem does not behave like that, you might have made a mistake and need to go back and look at it.

 Example 3

The distance in feet that a racehorse travels is given by the function

$$ d(t) = 2t^2 $$

Find d(t + h).

Find the difference quotient.

We will be substituting (t+h) for t in the expression 2t ^2.

$$ d(t+h) = 2(t+h)^2 $$

$$ 2(t^2 +2th + h^2) $$

This gives us (2(t+h).

$$ = 2t^2 + 4th + 2h^2 $$

Now lets calculate the difference quotient.

$$ \frac{d(t+h) - d(t)}{h} = \frac{2t^2 + 4th + 2h^2 -2t^2}{h} $$

Cancel stuff out and rewrite it.

$$ \frac{4th + 2h^2}{h} = \frac{h(4t + 2h)}{h} $$

Cancel those h’s. We get:

$$ 4t + 2h $$

If t=7 and h=.01, then the difference quotient becomes:

$$ 4t + 2h = 4(7) + 2(.1) = 28.2 $$

 So functions are very interesting. They can tell us a nice picture on what each one means. Constant and linear functions are easy to understand but its also important to remember what they represent and how they look graphically.

The slope on a graph is the degree of the rise and fall of a line between two points. This is also the rate of change. Rate of change tells us how fast any quantity changes. This then leads us to difference quotients.

A function is often used to model data that is important to us. It is useful in the sense that it can allow us to somewhat predict what will happen in the future mathematically.

Chapter 3 - Introduction

A function or mathematical model will not ever be exact. So a linear function is really just an approximation. With that said, linear functions and models of data can be very useful. You see them often in graphs. They are the straight lines you see representing data. A linear function usually takes the form \(f(x) = ax + b \). A line from this function on a graph can intersect one or both of the axes. Lines like this can have a slope. Slope is shown as :

$$ \frac{\Delta y}{\Delta x} $$.

If the line crosses the y-axis it is called the intercept. If we have a linear function \(f(x) = 2x + 5 \), 2 is the slope and 5 is the y-intercept.

The x-intercept is where the line crosses the x-axis. If we fill out the equation with intercepts and slope it should equal out to zero. This is how you can check your answer. The x-intercept is also called the <zero> of the function. If the slope of a function is not zero, then the graph only has one x-intercept.

Modeling Functions

Functions are often used to model data. When we have a graph of data, the slope also tells us its rate of change. This is useful if it is changing at a constant rate. The constant rate of change is the slope of a straight line on a graph. So basically, if the slope is the same between two points then you can use a linear function. An exact model will be able to describe the data very closely, if not exactly. We usually never see exact models of anything outside of a textbook so keep that in mind.

Linear Regression

The example I have mentioned have all been regression problems. This is because we have used one variable to predict the value of another variable. This is the very definition of linear regression.

Equation Of A Line

As discussed before, any quantity that increases at a constant rate, can be modeled with a straight line. The slope of this line can be computed easily. The \( \Delta y = y - y_1 \) and \(\Delta x = x - x_1 \) are what is needed. In fact:

\[ \frac{ \Delta y}{ \Delta x} \] will give us the slope itself. This is the slope formula. It also has other versions.

They look like this:

\[ m = \frac{ y - y_1}{x - x_1} \]

and

\[ y = m(x - x_1) + y_1 \]

Point-Slope Form

You will use the last two equations more than the first. The first is just a generalization. The latter two are just different forms but say the same thing. The last one is also known as the point-slope equation. So here is an example I found to show how this works.

Example

Find an equation of the line passing through the points (-2, -3) and (1,3).

You will start by finding the slope or rate of change. If you are looking at a graph, this will be the line you are looking at.

\[ m = \frac{3 - (-3)}{1 - (-2)} = \frac{6}{3} = 2 \]

That gives us a slope of 2 for this line. The line consists of the two points mentioned above. Now we just use one of the points along with our slope and put it in the point-slope form. I will use the first point given.

\[ y = 2(x + 2) - 3 \]

Slope-Intercept Form

The slope-intercept form is another variation. It is \( y = mx + b \). M is the slope and b is the y-intercept.

Example

Find a point-slope form for the line.

First we have a slope of \(-\frac{1}{2}\) passing through the point (-3, -7).

The point-slope form is \( y = m(x - x_1) + y_1 \). Just plug in the values that we were given. We get:

\[ y = -\frac{1}{2}(x +3) -7 \].

We get the slope-intercept form by just simplifying this expression.

\[ y = -\frac{1}{2}x - \frac{3}{2} -7 \]

\[ y = -\frac{1}{2}x - \frac{17}{2} \]

Intercepts

An equation of a line has another form which is called the standard form. It looks like: \( ax + by = c \). This is the third equation of a line we have talked about now. So we have in total: point-slope form, slope-intercept form, and now the standard form. To find the intercepts from a standard form equation, you will set y or x to zero and solve the equation. If y is set to zero then you will get the x intercept. Setting x to zero will then give you the y intercept.

Example

Find the intercepts for the equation \( 4x + 3y =6 \).

Start with setting y=0.

\( 4x + 0 = 6 \)

\( x = 6/4 \)

\( x = 3/2 \) = x-intercept

Now let us set x to zero to find the y intercept.

\( 0 + 3y = 6 \)

\( y = 6/3 \)

\( y = 2 \) = y-intercept

Other Types Of Lines

There are even more types of lines you can see on a graph. Don’t worry, they are all easy. The first is a horizontal line. On a graph, this represents a constant function. It is the easiest to recognize and do anything with. A horizontal line has the same y coordinate but different x coordinates. It also has a zero slope.

A vertical line is the opposite. It will have the same x coordinate but different y coordinates. Its slope will be undefined.

Parallel lines are the next type. They can’t be vertical to be considered parallel. These lines also need to have the exact same slope. If these conditions are met they will be considered parallel.

Perpendicular lines are the last type. For two lines to be perpendicular they need to be negative reciprocals.

Related Data

This may seem like common sense, but it is also important to state. Data can be related to each other. If one quantity changes then another will change because it can be directly related. However, it is important to be able to see if data changes because it is related or not.

Recapping It All

Point-slope form = \( y = m(x - x_1) + y_1 \). This is used to find the equation of a line if you have two points or one point and a slope to work with.

Slope-intercept form = \( y = mx + b \). This is a discrete equation for any certain line. It is obtained by using the slope and and the y-intercept.

Interpolation = Estimated values that are between two data points.

Extrapolation = Estimated values that are not between two data points.

In this section I covered more of the basics, which include line equations and working with data points. The next chapter will be over linear equations and working with them.

Chapter 4 - Solving Linear Equations

An equation is a statement that says two mathematical expressions are equal. Equations can have one or multiple variables. Solving an equation means finding all values for the variable that make the equation a true statement. The simplest type is the linear equation. A linear equation with one variable is an equation that can be written in the form: \( ax + b = 0 \).

Example

Solve the equation \( 3(x-4) = 2x - 1 \).

\[ 3x - 12 = 2x - 1 \]

\[ x = 11 \]

Example

Solve \( 3(2x - 5) = 10 - (x + 5) \).

\[ 6x -15 = 10 -x -5 \]

\[ 6x - 15 = 5 - x \]

\[ 7x = 20 \]

\[ x = \frac{20}{7} \]

When fractions or decimals are in an equation, you multiply each side of the equation by the least common denominator of all fractions in the equation.

Example

Solve: \( \frac{T-2}{4} - \frac{T}{3} = 5 - \frac{1}{12}(3 - T) \)

\[ 3(T-2) - 40T = 60 - (3 - T) \]

\[ 3T - 6 - 4T = 60 - 3 + T \]

\[ -2T = 63 \]

\[ T = \frac{-63}{2} \]

Example

Solve: \( .03(z - 3) - .5(2z + 1) = .23 \)

To eliminate decimals in this case, multiple them by 100.

\[ 3(z-3) - 50(2z+1) = 23 \]

\[ 3z - 9 - 100z - 50 = 23 \]

\[ -97z = 82 \]

\[ z = \frac{-82}{97} \]

The equation \(f(x)\) = \(g(x)\) results whenever the formulas for two functions (f) and (g) are set equal to each other. A solution to this equation corresponds to the x-coordinate of a point where the graph of (f) and (g) intersect.

Example

Solve: \( 2x - 1 = \frac{1x}{2} + 2 \)

\[ 2x = \frac{1}{2} x + 3 \]

\[ \frac{3}{2}x = 3 \]

\[ x = 2 \]

Linear equations and functions can be solved symbolically, graphically, and numerically. Symbolic solutions are always exact. Graphical and numerical solutions are approximated to some degree. The intermediate value property states that if two points are connected, then (f) assumes every value between the (y) points at least once.

Example

A survey found that 76% of bicycle riders do not wear helmets. Find a symbolic representation for a function that computes the number of people who do not wear helmets. Also, there are around 38.7 million riders who do not wear helmets. Write a linear equation that gives the total number of riders.

linear function that computes 76% of (f) is: \( f(x) = .76x \).

We must find the x-value for which \( f(x) = 38.7 \). The equation is \( .76x = 38.7 \) which evaluates to \( x = \frac{38.7}{.76} \) and is equal to 50.9 million riders.

Solving Problems

Read the problem and make sure you understand it. Assign a variable to what you are being asked to find. Write an equation that relates the quantities described in the problem. Solve the equation and determine the solution. Check your solution and make sure it seems plausible.

Example

A large pump can empty a tank of gasoline in 5 hours and a smaller pump can empty the same tank in 9 hours. If both pumps are used to empty the tank, how long will it take?

We are looking for time, so let time equal (T). In one hour the large pump will empty \(\frac{1}{5}\) of the tank and the smaller pump will empty \(\frac{1}{9}\) of the tank. The fraction of the tank they will empty together in 1 hour is \( \frac{1}{5} + \frac{1}{9} \). So, in (T) hours, the fraction of the tank that the two pumps can empty is \(\frac{T}{5} + \frac{T}{9} \). Since the tank is empty when this function reaches 1, we can use \(\frac{T}{5} + \frac{T}{9} = 1 \).

\[ \frac{T}{5} + \frac{T}{9} = 1 \]

\[ \frac{45T}{5} + \frac{45T}{9} = 45 \]

\[ 9T + 5T = 45 \]

\[ 14T = 45 \]

\[ T = \frac{45}{14} \]

\[ T = 3.21 hours \]

Example

In one hour an athlete travels 10.1 miles by running 8 mph and then at 11 mph. How long did the athlete run at each speed?

We are asked to find the time spent running at each rate. If we let (x) represent the time spent running at 8 mph, then 1-x represents the time running at 11 mph because the total running time was 1 hour.

Distance (d) equals rate times time (T) or \( d=rt\). In this example, we have two rates and two times. The total distance must sum to 10.1 miles.

\[ d = 10.1 = 8x + 11(1-x) \]

\[ 10.1 = 8x + 11 -11x \]

\[ 10.1 = 11-3x \]

\[ 3x = .9 \]

\[ x = .3 \]

So the athlete runs .3 of an hour at 8 mph and .7 of an hour at 11 mph.

Example

A person 6 feet tall stands 17 feet from the base of a streetlight. If the person’s shadow is 8 feet, estimate the height of the streetlight.

We are being asked to find the height of a streetlight, this will be (x).

You can use similar proportions here to solve this quickly.

\[ \frac{x}{6} = \frac{25}{8} \]

\[ x = \frac{(6)(25)}{8} \]

\[ x = 18.75 ft. \]

The streetlight is 18.75 feet tall.

Example

Pure water is being added to a 30% solution of 153 ml of hydrochloric acid. How much water should be added to reduce it to a 13% mixture?

We are asked to find the amount of water that should be added to 153ml of 30% acid to make it a 13% solution. Let this amount of water equal x. Then x + 153 equals the final volume of the 13% solution.

Since only water is added, the total amount of acid in the solution after adding the water must equal the amount of acid before the water is added. The volume of pure acid after the water is added equals 13% of x + 153ml, and the volume of pure acid before the water is added equals 30% of 153ml. Our equation becomes:

\[ .13(x + 153) = .30(153) \]

\[ x + 153 = \frac{.30(153)}{.13} \]

\[ x = \frac{.30(153) -153}{.13} \]

\[ x = 200.08 ml \]

In this chapter I have talked about the fundamental linear equation. It is the easiest to understand and algebra essentials. You can see how many quantities are modeled and then estimated using basic algebra.