How to find the equation of a line

If you know the intercepts.

Here is a sample problem:

you have the intercepts:

x intercept: 4
y intercept: -2

1. From here, we can find out two points:

(4, 0) and (0, -2)

Now, since we have two points, we must have two equations.

y = mx + b is the standard equation of a line.

2. Let's plug the values in for the x intercept.

y = mx + b
0 = m (4) + b
0 = 4m + b  

3. Now y intercept:

y = mx + b
-2 = m (0) + b
-2 = 0 + b
-2 = b

4. Since we found what b is in this equation, we can sub it into the first equation.

0 = m (4) + b,    b = -2     * m (4) is the same as 4m.

0 = 4m  -2
2 = 4m
2/4 = m
m = 2/4
m = 1/2
m = 0.5

5. Now we know what m and b are. We can form the equation of the line.

y = 0.5x -2

That is your answer!


Why can't you plug x and y intercept points into the same (ONE) equation? Because they change. They are not constant - x and y can be any number on the graph!

Only the values of m and b are constant. So that's why you don't write what y and x are (in numbers) in the equation!

Also, if the question asks for you to put the equation in standard form, Ax + By = C, it is very simple!

Shortcut to finding intercepts

Hello, the math nerd is back! lol. i know sorry, same topic. but this time, i have something cooler. I have a shortcut method to find the x and y intercepts!


General form of the equation of a line: y = mx + b

Here is the shortcut formula for finding the y intercept:  y = b.

meaning: x = 0, y = b.     Point: (0, b)

Proof:

y = mx + b. at the y intercept, x is 0. Let's sub that in.

y = m (0) + b
y = 0 + b
y = b

If there is a constant, a, infront of y, then the shortcut formula would be this: y = b/a.

Proof:
ay = mx + b
ay = m (0) + b
ay = 0 + b
ay = b
y = b/a

Your point would be (0, b/a).

X intercept: shortcut formula is X = -b/m

X = -b/m, y = 0  P: (-b/m, 0)

Proof:

y = mx + b. At the X intercept, y = 0. Sub it in.

0 = mx + b
-b = mx
-b/m = x
x = -b/m

This stays true even if there is a constant before y, since anything multiplied by 0 is still 0.

example: 2y = mx + b

2 (0) = mx + b
0 = mx + b
-b = mx
-b/m = x
x = -b/m


=)

and the most relaxed yet productive week of the year goes to......

THIS WEEK! YAY CONGRATSSSS =)

I have done the most work this week, and have felt equally relaxed!

YAY. Hurray for work and play balance!


Monday - Finished enough calc problems (quota), physics, english questions and french introduction! And then I had free time ...

Tuesday - finished almost all calc problems from the package! had lots of relax time =)

Wednesday - lots of relax time, finished all calc problems, doing physics now!

Thursday - no school (OSSLT)! yay. (30 min classes = not worth it) going to finish rough copy of french essay, catch up on chem, finish reading act 3 for shakespeare + questions! study for act 1/2 quiz tmr.

Friday - Catch up on chem / finish act 3.

strangely enough, i was able to focus on work and not get distracted! I also completed physics + chem homework for the first time at home when it wasn't before a test! YAY.  getting better at focusing. i think its my subconscious worrying over my dropping marks. updated marks going into unis on monday! eep! the only 90 i have is tech D=. ugh. i hope calc is good, and eng has gone up. so so so it should hopefully even out my droppage.

lalalalaala! also, for some good reason, i'm suddenly enthusiastic about math! =) yay. i think its from helping people, and the fact that i actually understand what i'm doing now! yay =) teaching = learning for both me and the "student". also, there are only a few people who are really good at teaching me math, so yay & thanks to them. =)

Intercepts

Definitions from dictionary.com:

1. to take, seize, or halt (someone or something on the way from one place to another); cut off from an intended destination: to intercept a messenger.

2. to see or overhear (a message, transmission, etc., meant for another): We intercepted the enemy's battle plan.

3. to stop or check (passage, travel, etc.): to intercept the traitor's escape.

4. Sports . to take possession of (a ball or puck) during an attempted pass by an opposing team.

5. to stop or interrupt the course, progress, or transmission of.

6. to destroy or disperse (enemy aircraft or a missile or missiles) in the air on the way to a target.

7. to stop the natural course of (light, water, etc.).

8. Mathematics . to mark off or include, as between two points or lines.

9. to intersect.

10. Obsolete . to prevent or cut off the operation or effect of.

11. Obsolete . to cut off from access, sight, etc.

–noun

12. an interception.

13. Mathematics .

a. an intercepted segment of a line.

b. (in a coordinate system) the distance from the origin to the point at which a curve or line intersects an axis.

Haha, I'm really starting to like this format. Plus, I learn new definitions every day!

Today we will be looking at the mathematics (you guessed it!) definition, and I will show you a sample problem.


Definition of an intercept for grade nine level math: Where a line (or line segment) intersects (meets) the x or y axis. The point that they meet at (x, 0) or (0, y), is called an intercept.
My friend asked me how to do this question today, so I will share with you too! This is grade nine level.

Exam Study Tip - Numbers One and Two

TIP NUMBER ONE:

I wish I thought of this earlier.. sigh... now I must find an alternative to study for finals.

1. At the very beginning of the course; whether it be the semester or the year, make a box (or find one) large enough to hold little snippets of paper, one for each subject that requires lots of memorization or calculations.

2. When you're assigned homework questions, copy down the ones that you had trouble with onto a piece of scrap paper. You can even put in definitions!

3. Locate the answer (text book, teacher) and make sure that it is 100% correct. Write it on the back of the question.

4. Fold and put it in the corresponding box.

5. When you're getting close to the exam date, randomly draw out questions from the box. Since most of these questions are questions that you've had trouble with in the past, they are probably the best questions to study from, to make sure that you still remember and understand how to do them. Also since you've collected them throughout the entire year (or term, whatever time frame fits best for you), you'll have a variety of questions, ranging from easy, medium to hard! To ensure you have a good mixture of easy, medium and hard questions, put in ANY and ALL questions that you either got wrong on the first try, or had trouble with! =) Remember, the exam will not just have hard questions, so don't neglect the "easy" ones, or they won't be so easy on the exam!


TIP NUMBER TWO:

1. Make a cheat sheet for each chapter or section of the book that you would bring into the test for that section/unit if you could. This will help you to remember important and neglected things from that chapter, formulas, and refresh your memory. Also, writing it down on paper helps you to learn/remember in two ways: sight, and motor memory!

2. Either enlarge and hang it on your wall, or somewhere you look at often - so that you'll commit these images to memory, and if you have a good spatial sense, that'll be your "photographic memory". Unless you actually do have photographic memory, then this should help! I remember most charts, graphs, and patterns by remembering exactly where things were on a poster, hand out, or chalkboard, especially if I've seen it more than once.

Optimization

Google's definitions:

Make the best or most effective use of (a situation, opportunity, or resource)

• to optimize viewing conditions the microscope should be correctly adjusted

Rearrange or rewrite data to improve efficiency of retrieval or processing

Let's look at the second one. In calculus, I am learning how to solve word problems that have to do with optimizing - finding the maximum or minimum values of something.

Here are the steps:

1. From the word problem, derive two equations (for my level anyway so far). This means two variables, usually one given constant and one unknown constant.

2. Rearrange the equation with the given constant so that one unknown is isolated.

3. Sub that equation into the second equation, so that there is only one variable in the "new" equation.

4. Now, since the "new" equation is equivalent to an unknown constant, we can take the derivative of the equation, which will give you the slope of the underived equation.

This is the tricky part that I didn't understand until today =)

Why do we take the derivative of the equation? To find the slope.

Why do we want to know what the slope is? - we don't (in some sense). we already know that the slope is zero at a maximum and minimum value for x, so we just need to find the unknown variable (x).  Don't get it? Take a look at this graph:

f(x) is the graph of the "new" equation, underived. (the one with both equations combined)

f '(x) is the graph of the derivative of the above equation. there are two because you can have a question asking for the maximum or the minimum.

As you should remember from previous years of doing math, the slope of a straight line, y = b, is always zero.  By looking for the slope of the tangent at a maximum or minimum point, we can have the equation with one variable on one side equal to zero, on the other side.    (ex.  4x+1 = 0)

Also, another reason why you can just equal your new equation to zero is because the unknown constant is a constant (pretty redundant, but still helpful to know). What does this mean? When you find the derivative of a constant, it is zero. Isn't that a coincidence? Not really, it's supposed to be zero, or else there'd be something wrong with this kind of math (also called calculus)!

Let's move on now...

5. With the derivative of the equation, we should now be able to solve for x. You should only have one unknown now (since the constant turned into zero).

6. If you got a quadratic equation, just factor it or use the quadratic equation if you can't (your scientific calculator should be able to do this too). If it was a monomial, then that's easy!

7. Now, to make sure that you have the right value (s), plug them into the equation you made before you took the derivative of it. If you have multiple values, choose the value for your variable (x) where f (x) = largest/smallest.

8. You can also use the sign chart to check your solution. (if it's negative, then you have a negative slope = going down,  positive = going up. going from left to right, a max value should have + going to -, and vice versa for min values)

9. If you got it right, YAY CELEBRATEEEEEEEEEEEEEEE!

If not, you probably did some funky algebra! Go back and check your steps (tedious task, but necessary!).

* this post is mainly for helping me understand and study.... and also good for me to look back at later when studying for final exam. I understand better and things "click" when I "teach" stuff. =) thanks for helping me learn by helping you learn!

Anti-acid!

So how to deal with heart burn... or reduce it......things I can do anyway... (other than taking an antacid like tums)

1. Don't drink coffee, or drink it while eating something, to absorb the acid.
2. Keep some sort of bread or baked good on me at all times. (to eat when experiencing heart burn)
3. Don't drink pop (not that I like carbonated drinks anyway).
4. Don't eat too many clementines at one time =( or tomatoes!
5. Eat breakfast!
6. Don't stuff myself @ dinner..and slow down on eating.
7. No late night snacking, or food after 10pm.
8. Don't lie down or sleep after eating.
9. Don't wear tight clothing lol...... or just don't wear anything!
10. Use the washroom when I have to! No holding it.

Hopefully heartburn isn't telling me that i have gastroesophagael reflux or.... that it's not being mistaken for angina or a heart attack .. but i dont have all the symptoms for heart attack so i should be okay! =D

reminder: to call the doctor about my physical/xray results.