「英語で悩むあなたのために」コミュの英語「で」学ぶ-----数学

　今私の家には高校生が２人同居しています。
　メイドさんのかわりのようなものなのですが、うちから学校に通わせつつ子供の世話などをしてもらっています。そして同時に私が彼女たちの家庭教師も勤めています。

　２人とも空き時間を使ってなかなか熱心に勉強しており、今日は幾何学の勉強をしました。
　レベルは決して高くありません。極めて基礎的な内容で日本なら中学生でもわかる内容かも知れません。

　日本の英語教育は基礎文法を学んだあと和訳に力点を置きすぎており、教材としてもすぐ「文学」に走る傾向がありますが、本来、そういう文学作品に入る前に「説明文」の読み書きをしっかり見につけるべきです。

　そういう意味から言って、「英語で書かれた教科書」を読む機会があると大変ためになります。
　つまり、「英語を学ぶ」次の段階では「英語で学ぶ」ことが大切だと感じます。

　彼女たちに数学を教えるため、あらかじめ私自身が教材を予習します。
　以下に今日学んだ内容を一部ですが紹介します：

　　　　　　Module 1
　　　　　　Plane Coordinate Geometry

★What this module is about

This module will explain to you the relationship among line on the plane. This will also tell you about intersecting lines, non-intersecting lines, characteristics of parallel lines and perpendicular line. Further more, this will tell you how to determine the intersection of two lines if there is any.

★What you are expected to learn

This module will help you
1, Determine the point of intersection of two line
2, Compute for the coordinates of the intersection of two lines.
3, Determine without graphing if the given lines are parallel, perpendicular or neither.
4, Define algebraically parallel and perpendicular lines.

★How much do you know

Given each pair of lines, determine if they are a) intersecting but not perpendicular, b) perpendicular and c) parallel

1. y= 3x - 7
y= 3x + 1

2. 2x + 3y = 5
3x - 2y = 9

3. x + 2y = 9
4x + 3y = 1

4. 3x - 2y = 3
3x + 5y = 14

5. 2x - 4y = 1
4x - 8y = 7

6. The slopes of parallel lines are ________________.

7. What is the slope of the line parallel to 2x + 4y -3 = 0?

8. What is the slope of the line perpendicular to x -6y +5 = 0?

9. Write equation of the line parallel to 5x + 8y = 7 and passing through (2, 4).

10. At what point do the line 3x - y = 7 and 5x + y = 9 intersect?

★What you will do

　　　　Lesson 1
　　　　Intersection of Two Lines

Lines on the plane can either intersect or not. If two lines intersect, then there is a common point between them. That common point is where the two lines intersect. In a plane, this point has two coordinates, the x and y coordinates. The coordinates of the intersection can be solved algebraically. In second year algebra, you are taught how to solve systems of linear equations. That knowledge will help you a lot in understanding this session. Since every equation of the line assumes the general form ax + by + c = o, to get the intersection of two lines, you solve for the value of x and y common to both equations.

To solve for the value of x and y algebraically, there are methods that can be used. The following steps can help you in solving systems of linear equations.

1, Eliminate one variable by using
a. addition or subtraction
b. finding the value of one variable in terms of the other variable.
2, Solve for the value of the other variable
3, Substitute the computed value of one variable on either of the two equations to solve for the value of the remaining variable.

Examples:

1, Find the point of intersection of the lines whose equations are 3x - y = 10 and 5x + y = 14.

Solution:
To find the intersection of the two lines, solve the system by eliminating one variable through addition. Add the similar terms of the first equation to that of the second equation.

Equation 1 3x - y = 10
Equation 2 5x + y = 14
------------
Add 8x + 0 = 24
8x = 24
x = 3

Then replace x with 3 in either Equation 1 or Equation 2

Equation 1 3x - y = 10
3(3) - y = 10
9 - y = 10
- y = -1
y = 1

Therefore the point of intersection of the two lines is ( 3, -1 )


2. Find the point of intersection of the two lines whose equations are x + 2y = 4 and 3x + 5y = -21.

Solution:

Another method of eliminating one variable is through substitution. You get the value of one variable in terms of the other variable.

Equation 1 x + 2y = 4
Equation 2 3x - 5y = -21

Using Equation 1, solve for x in terms of y

x + 2y = 4
x = 4 - 2y

Using Equation 2, replace x with 4 - 2y

Equation 2 3x - 5y = -21
3(4 - 2y) - 5y = -21
- 11y = -21 - 12
-11y = - 33
y = 3

Then replace y with 3 on either Equation 1 or Equation 2 to solve for the value of x.

Equation 1 x + 2y = 4
x + 2(3) = 4
x + 6 = 4
x = -2

So the intersection of the two lines is the point whose coordinates are ( -2, 3 ).


3. At what point do the lines 3x + 2y = 12 and 4x - 3y = -1 intersect?

To eliminate a variable in this example, first find equivalent equations with equal but opposite coefficient in x or in y by multiplying one or both equations by a number factor.

Equation 1 3x + 2y = 12
Equation 2 4x - 3y = -1

3(3x + 2y) = 3(12) >>> 9x + 6y = 36

2(4x - 3y) = 2(-1) >>> 8x - 6y = -2

Add Equation 1 and 2 17x + 0 = 34
7x = 34
x = 2

Then replace x with 2 in either Equation 1 or 2

Equation 1 3x + 2y = 12
3(2) + 2y = 12
6 + 2y = 12
2y = 6
y = 3

Therefore, the point of intersection is (2, 3).


Example 4. Find the point of intersection of the lines y = 2x - 2 and y - 2x = 1.

Solution: Using any of the previous methods, solve the system of equations.

Equation 1 y = 2x - 2
Equation 2 y - 2x = 1

Using Equation 1, solve for y in terms of x

y = 2x -2

Replace y by 2x - 2 in the second equation.
(2x - 2) - 2x = 1
2x - 2 - 2x = 1
-2 = 1 , this is a false statement.

Solving the system led to a false statement. Therefore the system is inconsistent and has no solution. So the lines do not intersect. There is a way of determining whether the two lines intersect or not. The following theorem can be used to determine if two lines intersect or not.

Theorem: If two non-vertical lines intersect, then their slopes are not equal.
This theorem can be verified as follows.

(グラフ：添付画像参照）

Let A, B and C be three distinct points on a plane. By line determination postulate, line AB and line AC can be constructed with A ans a common point or point of intersection.

Get the slopes of the two line.

For line AB, the slope

m1 = (y2 - y1)/(x2 - x1)

For line AC, the slope
m2 = (y3 - y1)/(x3 - x1)

Since the numerators and denominators in the two fractions are different, then the two slopes are not equal. Thus

m1 is not equal to m2

The converse of the theorem, "If two non-vertical lines have different slopes, then the lines are intersection," is also true.