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Dr. MathExplains Algebra
Learning Algebra Is Easy! Just Ask Dr. Math!
John Wiley & Sons
ISBN: 0-471-22555-X
Chapter One
Linear Equations Linear equations (equations that describe a line) are the simplest equations that mathematicians study. The work you've done with integers and coordinate graphing will help you with the topic of linear equations. If you have two points on a graph, you can connect them to form a line. When you see a line on a graph, you might notice some of the points it goes through. You could be precise in naming those points by identifying both the x- and y-coordinates. You might also notice if the line is slanted to the left or to the right or if it is horizontal or vertical. Both the location of the points on the line and the slant of the line are important characteristics used to describe a line.
To describe a line algebraically, you write an equation of the line. When you speak of those equations in general, you are referring to linear equations. Once you're familiar with some examples, you can tell a lot about what the graph of an equation will look like just by looking at the equation.
Because linear equations are the most simple examples of the equations that mathematicians study-and because they are so useful for describing things that go on in the world-mathematicians have developed many tricks and shortcuts for dealing with them. In this chapter, we'll learn some of those tricks and shortcuts.
In this part,Dr. Math explains:
linear expressions and equations
slope, intercepts, and slope-intercept form
graphing linear equations
1 Linear Expressions and Equations
Linear expressions and linear equations are related concepts. A linear expression is an expression with a variable in it; however, the variable is raised only to the first power. For example, 5a is a linear expression. Another example is 5a + 2. If instead you had 5a + 2 = 12, then you would have an example of a linear equation. What's important to remember is that an equation has an equal sign but an expression does not.
Linear Expressions
Dear Dr. Math,
What are linear expressions, and how will I use them in my life?
Arturo
Dear Arturo,
Thanks for your question. Let me start with the "what are they" part of your question.
I am going to explain mathematical expressions by comparing them to something you probably understand from studying English grammar in school. When we write, we use sentences to write a complete thought. In a sentence, there must be a noun and a verb, and often there are extras like descriptive words.
In mathematics, we also frequently write in sentences, but we use numbers and symbols to convey a thought. A complete mathematical sentence includes an equal sign or inequality sign (< or >) and at least one term on either side. For instance: 5 + 3 = 8 is a mathematical sentence, called an equation, while 9 < 100 is also a mathematical sentence, called an inequality.
When we write, we may also use phrases, which are groups of words that are not complete sentences, like "in a nutshell" or "good sport." In mathematics, we use phrases too, but they're called "expressions." Expressions can be just one number or several numbers and some symbols; however, there is no equal sign or inequality sign between them. For example, 8 is an expression, and so are 4/3 and 5 + 3.
Some mathematical expressions include letters that stand for something else. These letters are called variables because they represent numbers that can vary. Expressions with variables are used every day in all sorts of situations. Here is an example: Let's say that you have a car that can travel 15 miles for every gallon of gas in the tank. You could represent the total number of miles you can drive based on how much gas you have in the tank using the expression 15g, where g stands for the number of gallons in the tank. You can figure out how far you will be able to go by replacing g with the number of gallons in your tank.
Now for the linear part. A linear expression is an expression with a variable in it, but there is a special condition involving exponents.
You may not have learned about exponents yet, so I'll give a brief explanation. We use exponents to symbolize many multiplication operations using the same number. For instance, perhaps you must multiply 3 by 3 by 3 by 3 by 3. We would write this as [3.sup.5], which means the product of five 3's. We read the expression [3.sup.5] as "3 raised to the fifth power." Any number raised to one is simply that number: [4.sup.1] = 4; 5[a.sup.1] = 5a, and so on. We usually omit writing the "1" when a number is raised to the first power.
A linear expression is an expression with a variable in it; but only when the variable is raised to the first power. For example, 5a is a linear expression, because it is understood that a is raised to the first power, but 9[t.sup.2] + 8 is not a linear expression because t is raised to the second power.
There are many possible examples of linear expressions that you might use in your life. Another common one is computing how much money you might earn in a week working at a restaurant where you were paid by the hour. If your salary is $6 per hour, your total pay for a week could be expressed as 6h, where h represents the number of hours you worked during that week. Can you think of more examples now? -Dr. Math, The Math Forum
Equation, Function, or Formula?
Dear Dr. Math,
I keep hearing the words "equation," "function," and "formula" in my algebra class, and sometimes it seems as though they all mean the same thing. Am I missing something?
Aimee
Dear Aimee,
Good question! It's not something that gets discussed as much as it should, so I'm glad you asked about it.
An equation is simply an assertion that two quantities are equal, for example,
3 + 5 = 8
When we include a variable in an equation, for example,
3 + x = 8
we're still asserting that two quantities are equal, but now we're doing something more. We're also asserting that there is some value (or set of values) that can be assigned to the variable to make the equation true. In the equation above, only the assignment x = 5 will make the equation true.
When we include a second variable, for example,
3 + y = 2x
we're asserting that there are pairs of assignments that make the equation true. That is, we can choose a value for one of the variables, and that choice will determine a value (or values) for the other variable.
For example, suppose we choose to assign x = 5 in the equation above. Then we have:
x = 5: 3 + y = 2(5)
which is true only when y = 7. So the pair of assignments (x = 5, y = 7) is one solution to the equation. But there can be other solutions as well. For example, if we choose to assign y = 5, then we have:
y = 5: 3 + 5 = 2x
which is true only when x = 4. So (x = 4, y = 5) is another solution to the equation.
Note that we can manipulate the equation to look like this:
y = 2x - 3
This has the same meaning, but it's somewhat easier to work with. Now if we choose a value for x, we can evaluate the resulting expression, and that gives us the corresponding value for y:
Choose Evaluate Solution
x = 1 y = 2(1) - 3 = -1 (x = 1, y = -1) x = 2 y = 2(2) - 3 = 1 (x = 2, y = 1) x = 3 y = 2(3) - 3 = 3 (x = 3, y = 3)
When we write the equation in this way, we say that y is a function of x. That is, it's a kind of rule for starting with one value (the "input") and finding a matching value (the "output"). Often, when we turn an equation into a function, we'll simply drop the output variable and use a slightly different notation. Instead of
y = 2x - 3
we'll write
f(x) = 2x - 3
But it means the same thing.
So we've talked about equations and about functions. What about formulas? A formula is simply an equation that is so useful that we want to share it with other people. It's often written in the form of a function, although it's not restricted to that use. For example, the formula for the volume of a cylinder is:
V = [pi][r.sup.2]h V = volume, r = radius, h = height
Written this way, it tells us volume as a function of radius and height. But sometimes we already know the volume and the radius, and we need to find the height! We don't bother to make up a new formula, because we can just change this one around to find what we want. So a formula is often a function, but it doesn't have to be, and we don't have to use it that way. Mainly a formula is an equation that is useful enough to write down in a permanent location (like a book or a Web site), so that we can look it up instead of having to figure it out from scratch each time we want to use it. -Dr. Math, The Math Forum
2 Slope, Intercepts, and Slope-Intercept Form
We talk about "ski slopes" and how steeply a roof "slopes" when we're referring to something that is not horizontal or vertical but at a slant. In math we use the term slope similarly. It is a number that tells how steeply a line slants as it goes up or down. Slope is important, because if you know the slope as well as the location of any point on a line, you have everything you need to know to find all of the other points on the line. So often, when you're given some information about a line, the first thing you want to do is figure out what the slope is.
When we talk about intercepts, it makes sense to to look at a graph. In the Cartesian system of graphing, we have the x-axis and the y-axis. When we graph a line, it may cross the x-axis or the y-axis, or both. Any point at which it crosses an axis is called an intercept. The point where it crosses the x-axis is known as the x-intercept, and the point will be in the form (x, 0). The point where it crosses the y-axis is known as the y-intercept, and the point will be in the form, (0, y).
You can see how useful slope and intercepts are for describing or graphing a line. Because of this, one of the standard forms for linear equations shows slope and y-intercept clearly. It's called the slope-intercept form, and it's written:
y = mx + b
In this equation, the m stands for the slope of the line, and the b stands for its y-intercept. So this line has slope m and crosses the y-axis at (0, b).
What Is Slope?
Dear Dr. Math,
I am confused about slope. I'm not sure what it means, and I don't know how to use it to draw a line. Can you help?
Arturo
Dear Arturo,
The slope is just a number that tells how steeply a line goes up or down. If the line is perfectly level (it doesn't go up or down at all), the slope is zero. If, as you go to the right, the line gets higher, we say it is sloping up, or has a positive slope. If it goes down as we go to the right, we call that a negative slope.
So that's how to think of slopes. Here's how to measure (or draw) them. Let's first look at specific examples.
Suppose the slope is 1. That means that if you go 1 unit to the right, the line goes up by 1 unit. The units can be whatever you choose, so if you go 1 foot to the right, the line goes up 1 foot. If you go 1 centimeter to the right, the line goes up 1 centimeter, and so on. If you draw this line, you'll find it goes up at a 45-degree angle.
If the slope is 3, the line goes up more steeply. If you go 1 unit to the right, the line goes up 3 units, where "unit" can be inch, foot, centimeter, or whatever.
If the slope is 1/2, it means that if you go 1 unit to the right, the line goes up 1/2 unit. If the slope is 1,000, the line is very steep-going 1 unit to the right, the line rises by 1,000 units, and so on.
If the slope is negative, it works the same way, except the line goes down to the right. A slope of -1 means that going 1 unit to the right, the line drops 1 unit. A slope of -1/3 means for every unit the line goes to the right, it drops by 1/3 of a unit, and so on.
There's one nasty problem and that concerns lines that go straight up and down-you can't assign a sensible slope to them, because they never go to the right. We call these slopes undefined.
Often you'll get problems like this: What's the slope of a line that goes up 3 units for every 2 units it moves to the right? To get the answer, you just divide the motion up by the motion to the right. That means it goes up 1.5 units for each single unit to the right, so the slope is 1.5. Similarly, you might be asked for the slope of a line that goes up 3 units for every 2 units it moves to the left. To get that answer, you divide the motion up by the motion to the left. That means it goes up 1.5 units for each single unit to the left-which is the same as saying it goes down 1.5 units for each single unit to the right, so the slope is -1.5. -Dr. Math, The Math Forum
Point-Slope Equations
Dear Dr. Math,
I don't understand how to do point-slope equations. Can you explain them?
Sincerely,
Aimee
Dear Aimee,
The point-slope form of an equation is just one of many ways to write the equation of a line. It's handy to use when you know the slope of a line and one point on it.
A point-slope equation looks like this:
y - [y.sub.1] = m(x - [x.sub.1])
where m is the slope and [x.sub.1] and [y.sub.1] correspond to a point on the line.
In order to solve a problem (that is, write an equation of a line) using the point-slope equation, you need two things: a point on the line ([x.sub.1], [y.sub.1]) and the slope of the line.
For example, to find the equation of a line with a slope of 2 and a point on the line (-1, 3), m would be equal to 2, [x.sub.1] would be -1, and [y.sub.1] would be 3.
Plugging them into the point-slope equation, you get:
y - 3 = 2(x - (-1))
Then solve for y to simplify the equation.
Continues...
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