Expert: Paul Klarreich - 3/12/2007

Question
Hello,my name is Ciera and I have been looking all through my book to find out how to do these problems.I wasn't here for the teacher to explain that part of the chapter because I just moved here.The problems that I need to learn how to solve are problems like these:
y=1/2x-1 and y=-x+3.The thing that we are supposed to be learning about is called "graphing a linear equation."
Thanks for your help and I hope that it's not too much trouble:)
-Ciera
P.S.This is my first time here so I don't know if you do or not but could you also explain how to do the problem?Thanks again!

Answer
Questioner:   Ciera
Category:  Advanced Math
 
Subject:  Could you help me with these problems if it's not too much trouble?
Question:  Hello,my name is Ciera and I have been looking all through my book to find out how to do these problems.I wasn't here for the teacher to explain that part of the chapter because I just moved here.The problems that I need to learn how to solve are problems like these:
y=1/2x-1 and y=-x+3.The thing that we are supposed to be learning about is called "graphing a linear equation."
Thanks for your help and I hope that it's not too much trouble:)
-Ciera
P.S.This is my first time here so I don't know if you do or not but could you also explain how to do the problem?Thanks again!
...................................................
Hi, Ciera,

Some years ago Oldsmobile put out a midsize car with your name on it.  I hope your parents didn't name you after that car.  It's a nice name but it was a lousy car and you would have a heavy burden to bear.

Well, enough reminiscing and on to some math.

If you are learning how to graph linear equations, assuming you are doing it with paper and pencil, then there is good news and bad news:

Bad:   There is a certain amount of work involved.  These take some time.
Good:  The work is very straightforward.  As long as you are careful and well-organized, it won't be hard.

I'll show you one example, then look at the ones you sent.
......................................
Ex: Draw the graph of  y = 2x - 3.

Process: Determine two ordered pairs (x,y) that satisfy the equation. [A third one, as a check on your arithmetic, never hurts.]

To determine each ordered pair:

A. pick a value of x.  Yes, just pick one.  True, some work out easier than others, and with some experience, you will tend to pick those first.

B. Substitute that value of x into the equation.
C. Solve or simplify to get a value of y.
D. Write your result (x,y)

Do A,B,C,D at least twice.

Here we go:

Pick  x = 0.  [I like zero.  Some day you will, too.]
Subst: y = 2(0) - 3  << Use parentheses when substituting.
Simplify:  y = 0 - 3,  or  y = -3.
Write:  (0,-3)  << Don't be lazy -- do this!


Pick  x = 2.  [It's good to pick the second number not too close to the first.]
Substitute:  y = 2(2) - 3
Simplify:  y = 4 - 3,  or  y = 1.
Write:  (2,1)

Bonus:  Pick x = 5.
Substitute:  y = 2(5) - 3
Simplify:  y = 10 - 3,  or  y = 7.
Write:  (5,7)

OK, we have three ordered pairs.  That should give us three points, even though we only needed two.

This is the part I can't do for you -- drawing the graph.  You do this:

A. Get some graph paper. (Use a word processor or something to make some.)
B. Draw a set of axes, labeled x and y.
C. Mark some numbers on the axes to set your scale.
D1. Mark  (0,-3)
D2. Mark  (2,1)
D3. Mark  (5,7)

Now take your ruler and draw the line through those three points.  Make the line extend beyond the points.  Put little arrows at each end.  [You say the ruler doesn't line up all three?  Then you made some mistake -- that's why the third point is a check point.]

Finally, write the equation as a label on the graph with a nice curvy arrow pointing to the graph.  Make it look nice and people will think you enjoy it.

..........................................................

Now to the examples you sent:

y=1/2x-1.

I assume this means:

y = (1/2)x - 1

In the future, you must be careful about your notation, especially when sending a question.  When in doubt, PARENTHESIZE.

Since we have to take half of each x-value, I think I will choose only even values for x.  I don't love fractions -- I'm good at them, but I don't love them.

Pick  x = 0.  [I still like zero.]
Substitute:  y = (1/2)(0) - 1  
Simplify:  y = 0 - 1,  or  y = -1.
Write:  (0,-1)


Pick  x = 4.
Substitute:  y = (1/2)(4) - 1  
Simplify:  y = 2 - 1,  or  y = 1.
Write:  (4,1)


Pick  x = 6.
Substitute:  y = (1/2)(6) - 1  
Simplify:  y = 3 - 1,  or  y = 2.
Write:  (3,2)

OK. The rest I leave to you -- the marking of the axes, points, line, etc.
..........................................

y = - x + 3    << space things out when you send them.

That minus in front of the x causes trouble, but we will parenthesize carefully and avoid it.  Remember this concept:

When you see a minus symbol in front of parentheses it means 'the opposite of ()'.  In fact, we'll play with some negative numbers in this example.


Pick  x = 0.  [Even now, I still like zero.]
Substitute:  y = -(0) + 3  
Simplify:  y = 0 + 3,  or  y = 3.
Write:  (0,3)

Pick  x = 4.
Substitute:  y = -(4) + 3  
Simplify:  y = - 4 + 3,  or  y = -1.
Write:  (4,-1)

Pick  x = -4.  << as promised.
Substitute:  y = -(-4) + 3  
Simplify:  y = + 4 + 3,  or  y = 7.
Write:  (-4,7)

You can handle the rest.
.......................................
Of course there is more to it, and your teacher might like you to use a graphing calculator or program.  

But this should get you started.