Introduction to Linear Equations and Slope/Intercept Form
Date: 03/19/2005 at 02:36:01 From: Vi Subject: A curious question I have about equations Dear Dr Math, Hi! I am in Year 8 in Australia. I was looking through your middle school archives (which is equivalent to junior high school here) and I even tried the high school ones, even though I won't be in Year 9 until next year :) In the algebra section, you talk of 'linear equations'. Can you explain to me what they are? I've never heard my teachers talk about them, and the answers aren't really clear to me. Can you help, step by step, in an easy way that doesn't baffle the brain? Once I get it. I could probably solve them easily, as algebra isn't that hard for me! Thanks for any replies!
Date: 03/19/2005 at 11:43:54 From: Doctor Ian Subject: Re: A curious question I have about equations Hi Vi, When you first get into algebra, you start with equations where you have one 'unknown', or 'variable'. That might look like 3x + 2 = 5 '3x' means '3 times the quantity x' With an equation like this, your job is to find the value of x that makes the equation true, e.g., 3*1 + 2 = 5 so x = 1 is the value of x that makes it true. Sometimes you can have more than one value, e.g., x^2 - 1 = 3 'x^2' means 'x squared' or 'x*x' where x = 2 and x = -2 both make the equation true. So far, so good? Eventually, you add a second variable, to get equations that look like y = 3x + 2 Now there is no longer a limited number of values that make the equation true. Why? Suppose I _decide_ that x is 1. Then y = 3*1 + 2 y = 5 and if I make y equal to 5, the equation is true. But if I decide that x is 2, then y = 3*2 + 2 y = 8 and if I make y equal to 8, the equation is also true. I think you can see with a little thought that I can give x any value I want, and I'll always be able to come up with a value for y that 'saves' the equation. So the solutions to an equation like this are pairs of related (x,y) values. The two solutions we just found are (x=1,y=5) and (x=2,y=8). By convention, we always write them in (x,y) order, so we can just abbreviate those as (1,5) and (2,8). Now, here's the really interesting part. If I set up a coordinate plane with two axes, one for x and one for y, then I can place a mark at the location (1,5) by starting at the origin (where the axes cross), moving 1 unit to the right, and 5 units up: | - | - | - | - | - * (1 over, 5 up) | - | - | - | - | --|--|--|--+--|--|--|--| | - | - | And I can plot the point (2,8) the same way: | - | - * (2 over, 8 up) | - | - | - * (1 over, 5 up) | - | - | - | - | --|--|--|--+--|--|--|--| | - | - | Now, let's stop and ask a question: What if I plot _all_ the possible solutions? At first, that seems kind of silly. I wouldn't have time to figure them all out! It would take forever. But here's what's very, very cool. Simply by looking at the form of an equation, I can say a lot about what the plot of all the solutions would look like. For example, looking at our equation y = 3x + 2 I can say that all the solutions will lie on a straight line. That is, if I place a ruler over the two points we've plotted so far, and draw a line through the points that extends forever in each direction, every solution to the equation will be a point on that line. We say that the line is the graph of the equation. An equation like that, we call 'linear'. :^D There are other kinds of equations. For example, an equation like y = (x - 3)^2 + 4 is always going to be the equation of a parabola. The equation (x - 2)^2 + (y - 1)^2 = 4^2 is always going to be a circle. And so on. So a lot of what you'll be doing in algebra is learning how to look at an equation and know something about what its graph will look like. You also learn how to look at a graph and know something about what the corresponding equation will be. For example, with linear equations, you'll learn that you can describe any line using a quantity called 'slope' (which tells you how steeply the line ascends or descends as it goes to the right), and another quantity called the 'y_intercept' (which tells you the value of x where the line crosses the y-axis). If you know those two values, you can just write down the equation of the line: y = slope*x + y_intercept In our case, y = 3x + 2 the slope is 3 (that is, for each step to the right, we move three units up), and the y_intercept is 2 (if you extend the line, you'll see that it crosses the y-axis at y=2). Going in the other direction, if I know the slope and the y_intercept, I can plot one point at (0, y_intercept) and a second point at (0+1, y_intercept+slope) and that gives me the two points I need to fill in the whole line. I think you might be getting a sense for how easy it is to work with lines, compared to other kinds of graphs we might see. This is why we place a lot of emphasis on linear equations. In fact, they're so nice to work with that in the real world, even when we know that something is too complicated to be described using lines, we'll pretend that it's simpler than it is, just so the math works out nicely. (This is the same sort of idea that you use in learning to estimate 39*48 as 40*50--you give up some accuracy, but you also avoid a lot of work. It's a trade-off that occurs at all levels of mathematics!) Take a look at what I've written above, and let me know if it's clear. Or if any parts aren't clear, tell me what they are, and we'll keep going until we get an explanation you can follow. Then we'll put it in our archive, and other students who are in your situation can benefit from it too, okay? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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