Systems of two linear equations1 in two unknowns

Consider the equation

x — 2y = 5 (1)

In this equation x=7 and y=1, but also x=5 and y=0. There are many such pairs of values which satisfy the equation (1). To find pairs other than those given, choose a value of one letter, say y arbitrary, and then from (1) find the сcorresponding value of x. For example, let y=3. Then from (1)

x= 5 + 2y (2) whence x=5+2×3, x—5+6=11 and the pair of values x=11, y=3 satisfies the equation (1). The method for finding the pair of values satisfying both equations indicated above usually applies to pairs of equations of the form:

а₁x+ b₁у = c₁ (3)
а₂х + b₂у = c₂

where a_1, a_2, b_1, b₂, c_1, c₂ are known, and x and y are unknown quantities.

The equations (3) are termed linear because the unknown x and y enter to the first power only.

To solve a system of two linear equations in two unknowns, solve for one unknown in one equation and "substitute this result in the other equation, thus obtaining one equation in one unknown.

An alternative way² of solving a system of two linear equations, which is usually more convenient, is given by the following rule: multiply the two equations with numerical factors which are chosen so that³ the coefficient of one of the two unknowns have the same numerical values in both equations.

By adding or subtracting the two equations, a new equation with only one unknown quantity is obtained. Solve this equation. In order to find⁴ the second unknown quantity, substitute the value which has been found and solve for the remaining unknown quantity. An alternative method for finding the second unknown is to repeat the above process of finding the equal coefficient for the other unknown.

Notes:

¹ equations in two unknowns — рівняння з двома невідомими

² an alternative way — інший спосіб

³ so that — так що; таким чином, що

⁴ in order to find — для того, щоб знайти

EXERCISES

I.Read the following words paying attention to the pronunciation:

pair, where, there, compare, factor, letter, order, other, consider, enter, contain, obtain.

II.Underline all the suffixes and state to what part of speech the words belong:

equation, arbitrary, usually, convenient, corresponding, linear, equally, choosing, alternative, coefficient, numerical, system, factor.

III.Make up sentences of your own using the words and expressions given below:

in one unknown, in two unknowns, in three unknowns, to satisfy the equation, the method for finding, to obtain an equation, to establish.

IV. Answer the following questions:

1. What equations are termed linear? 2. What is the first operation in solving a system of two linear equations in two unknowns? 3. What do you obtain by adding or subtracting the two equations? 4. What operation do you perform to find the second unknown quantity?

V. Translate into Ukrainian:

In order to solve two equations in two unknowns, it is necessary to eliminate one of the unknowns by combining the two equations into one equation, which only contains one of the unknowns. This simple equation is then readily solved for that unknown in the usual way. With one of the original unknowns now known, its value can be substituted for the symbol in one of the equations, and from the resulting simple equation, the other unknown can be found. There
are several methods of eliminating one of the unknowns and combining the two original equations into one.

IV.Translate into English:

Рівнянням називається рівність, у якій одне або декілька чисел, позначених буквами, є невідомими.

Нехай, наприклад, сказано, що сума квадратів двох невідомих чисел х і у дорівнює 7; це можна записати за допомогою наступного рівняння з двома невідомими:

x² + у² = 7.

Рівнянням першого ступеню з двома невідомими називається рівняння типу

ах + bу = с,

де х і у — невідомі, а і b (коефіцієнти при невідомих) — задані числа, що не дорівнюють обидва нулю, с (вільний член— absolute term) — любе задане число.

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SQUARES AND SQUARE ROOTS

To square a number¹, you have learned, you must multiply that number by itself. The square root of a number is just the opposite. When you find the square root of a number, you are finding what number multiplied by itself gives you the number you began with². The sign for the square root is √. Thus, the square root of 25 is represented by √25. 25 is a perfect square. That is, a whole number (5) multiplied by itself will give you 25. Most numbers are not perfect squares. In that case, to get the square root of a number we may either find it by taking an arithmetic square root or by using a table.

The process of finding a root is known as evolution; it is the inverse of involution, because by the aid of this process we try to find that which is given only when raising a number to a power (viz. the base of the power), while the data given is just what is sought for³ raising a number to a power (viz. the power itself). Therefore the accuracy of the root taken may always be checked by raising the number to the power⁴. For instance, in order to check the equality: ³√125=5, it is sufficient to cube 5; obtaining the quantity under the radical sign, we conclude that the cube root of 125 has been found correctly.

Notes:

¹ to square a number — щоб піднести число в квадрат

² the number you began with — тут: початкове число

³ what is sought for — те, що шукається

⁴ by raising the number to the power — піднесенням числа в ступінь

EXERCISES

I.Read the following words paying attention to the pronunciation:

inverse, learn, perfect, order, for, opposite, not, must, number, thus.

II.Form Participles using the following verbs:

to square, to use, to raise, to multiply, to find, to check, to give, to begin, to obtain, to get, to take, to be.

III.Make up sentences of your own using the words and expressions given below:

to raise to power, to obtain the quantity, to square the number, to take an arithmetic square root, to use a table may be checked, conclude.

IV. Answer the following questions:

1. What operation should be performed to square a number? 2. What is a perfect square? 3. What do we do to get the square root of a number? 4. What is the process of finding a root called? 5. How do we check the accuracy of a root?

IV.Translate into Ukrainian:

Tables of squares are used by architects and engineers in working with squares of number. If you have a table of squares, you can find the approximate square root of any number. Sometimes it is not easy to find a square root by

inspection. If a table of squares is not at hand another method may be used.

V.Translate into English:

Щоб піднести в квадрат число, треба помножити це число на саме себе. Добування квадратного кореня — це дія, обернена до піднесення в квадрат. Щоб отримати квадратний корінь числа, ми можемо скористатися спеціальною таблицею. Правильність вилучення квадратного кореня можна перевірити, піднісши в квадрат отриманий результат; якщо вийде дане число, то корінь знайдено правильно.

TEXT 9

LOGARITHMS

An important step toward the lessening of the labour of computations was made in the seventeenth century by the discovery of logarithms. Logarithms permit us to replace long process of multiplication with simple addition; the operation of division with that of subtraction; the task of raising to any power with an easy multiplication; and extraction of any root is reduced to a single division.

The logarithm of a given number to a given base is the exponent of the power to which this base must be raised in order to obtain the given number.

Logarithms are exponents.

If a^x=b, the exponent is said to be the logarithm¹ of b to the base a, which we write x=log_ab.

The logarithm of a number to a given base is the exponent to which the base must be raised to yield the number.

Any positive number different from unity can be used as the base of a system of real logarithms.

Examples: If 10³ = 1000, then 3 = log₁₀1000

If 2³ = 8, then 3 = log₂8

If 5² = 25, then 2 = log₅25

The logarithmic system using 10 as a base is known as² the common or Griggs system and makes use of the fact that every positive number can be expressed as a power of 10. Since our number system uses 10 for a base, it is desirable for us to use 10 for the base of logarithms.

The following table shows the relationships between the exponential and logarithmic forms:

10³ = 1000 log₁₀1000 = 3.0000

10² = 100 log₁₀100 = 2.0000

10¹= 10 log₁₀10 = 1.0000

10° =1 log₁₀l = 0.0000

10^-1 = 0.l log₁₀0.1 = 1 or -1 or 9.0000-10

10^-2 = 0.01 log₁₀0.01 =2 or -2 or 8.0000-10

From this table it is clear that any number between 100 and 1000 is a power of 10 for which the exponent is greater than 2 but less than 3, and consequently, its logarithm is between 2 and 3 (2+a decimal). Similarly, the logarithm of 30 is (1+a decimal).

Later, when we use the table, we shall find log 30= 1.47712 which also means 10^1.47712=30. The positive decimal part of logarithm is called the mantissa and the integral part is called the characteristic.

Example. If log₁₀300=2.47712; 2 is the characteristic and 47712 is the mantissa.

Notes:

¹ the exponent is said to be the logarithm – кажуть, що показник ступеня – це логарифм

² is known as — відома як

EXERCISES

I. Read the following words paying attention to the pronunciation:

coefficient, exponent, except, logarithmic, yield, application, multiplication, transformation.

II. Form adverbs and adjectives using the following suffixes and translate the newly formed words into English:

-ly: part, simple, easy, real, common, clear

-less: use, number, power

III. Answer the following questions:

1. When was the discovery of logarithms made? 2. Why was the discovery of logarithms an important step? 3 What operations can be replaced by logarithms? 4. What logarithmic system is known as the common or Griggs system? 5. What fact does the common logarithmic system make use of?

V.Translate into Ukrainian:

In giving logarithm of a number, the base must always be specified unless it is understood from the beginning that in any discussion a certain number is to be used as base for all logarithms. Any real number except 1 may be used as base, but we shall see later that in applications of logarithms only two bases are in common use.

Suppose the logarithm of a number in one system is known and it is desired to find the logarithm of the same number in some other system. This means that the logarithm of the number is taken with respect to two bases. It is sometimes

It is important to be able to calculate one logarithms when the other in known.

TEXT 10

THE SLIDE-RULE

The slide rule presents one of the quickest and easiest ways of performing the operations of multiplication, division, raising to a power, and extracting a root, when not more than the first three or four digits of the result are necessary. Both straight and circular slide rules are in common use, but it is generally easier to learn to use the straight slide rule first. We represent here the fundamental principles of the straight (10 inches) slide rule as they are used in multiplication, division, reciprocals, proportion, squaring and extracting square roots.

The straight slide rule is composed of many bodies or "frames", a slide or "slip stick", and an indicator. The slide is the main body. The indicator is the glass containing the hairline and is used to locate numbers in the various scales.

The scales on a slide rule are graduated according to the mantissa of the positive real numbers, or, according to the logarithms of numbers from 1 to 10.

The top scale is divided into ten equal lengths (1 inch each for the 10-inch slide rule). The numbers from 1 to 10 on the bottom scale are located so that they correspond to their logarithms on the top scale. Note that the lengths between the numbers on the bottom scale decreases in size as the numbers become larger. The bottom scale is called a logarithm scale.

EXERCISES

I.Read, the following words paying attention to the pronunciation:

slip, stick, ways, raise, indicating, according, corresponding, locate, scales.

II.Give the following nouns in the singular:

rules, glasses, inches, principles, reciprocals, bodies, mantissas, lengths.

III.Make up sentences of your own using the words and expressions given below:

the 10-inch slide-rule, the quickest way of performing the operation, the easiest way of performing the operation, to graduate a scale, to locate numbers, a slide.

IV.Answer the following questions:

1. What operations can we perform with the help of the slide rule? 2. What kind of slide rules do you know? 3. What is the straight slide rule composed of? 4. How are the scales on a slide rule graduated? 5. What scale is called a logarithm scale?

PART III. GEOMETRY

Engineers, architects and people of many other professions use lines and figures in their daily work. The study of lines and closed figures made by lines is called geometry. Geometry is the branch of mathematics which investigates the relations, properties, and measurement of solids, surfaces, lines and angles.

TEXT 1

POINTS AND LINES

A point has no length, width or thickness. It merely indi­cates position. To represent a point in geometry we mark a dot and label it with a capital letter. For example, A or -A would be called "point A".

A line has no width or thickness. It has length and direction. An infi­nite number of straight lines can be drawn through one point.

Since a line extends indefinite in either direction, we must with line segments, or portions in lines. The segment is represented by two capital letters, one placed at each end. The line segment AB or BA is shown in Fig. 3. It can also be represented by small letters.

A line joins two points. Only one straight line can be drawn between two points. There are three kinds of lines: straight, curved and broken.

In Fig. 3 AB is a straight line; CD is a curved line; EF is a broken line. Notice that the lines are labeled by capital letters placed at the end of the line.

Lines that extend from left to right as the horizon are called horizontal lines. Examples of horizontal lines are lines on writing paper and all level lines which we find in man-made structures.¹

Note:

¹ man-made structures — спорудження, створені руками людини

EXERCISES

I. Read the following words paying attention to the pronunciation:

length, thickness, width, thin, straight, draw, through, curve, there, that, position, representation, profession, either, or, more.

II. Form nouns adding suffixes and translate the newly formed words into Russian:

-tion: construct, represent, multiply, form;

-ment: displace, measure;

-ing: draw, study, find.

III. Make up sentences of your own using the words and ex­pressions given below:

daily work, closed figures, to be represented by, thickness to extend, indefinitely, labeled by, a broken line.

IV. Answer the following questions:

1. What is geometry? 2. What are the characteristic fea­tures of a point? 3. How do We represent a point in geometry? 4. How many lines can be drawn through one point? 5. What is a segment? 6. How many lines can be drawn between two points? 7. What kind of lines do you know?

V. Translate into Ukrainian:

Geometry is the branch of mathematics which investi­gates the relations, properties and measurements of solids, surfaces, lines and angles.

The two points may be at any distance apart, so a straight line may be considered as having any length.

A broken line is a line formed of successive sections, or segments, of straight lines.

A curved line, or simply a curve, is a line no portion of which is straight.

VI. Translate into English:

Через будь-які дві точки можна провести пряму і при цьому тільки одну. Якщо на площині взяти будь-які дві точки і провести через них пряму лінію, то всі точки цієї прямої будуть знаходитися у цій площині.

TEXT 2

ANGLES

Measuring Angles. An angle is formed when two straight lines meet at a point. The lines are called the sides of an angle. The point at which the sides meet is called the vertex of the angle. The angle is read as angle BAC or CAB (Fig. 4).

The size of an angle depends upon¹ the amount one side has turned away from² the other. The length of the sides of an angle does not determine its size.

The unit of measure used in measuring an angle is the de­gree. A degree is a unit that equals 1/90 of a right angle and 1/360 of a circle. A right angle, therefore, contains 90 degrees (90°), and a circle contains 360 degrees (360°). The size of an angle is the number of degrees through which one side of the angle has turned away from the other side.

Kinds of Angles.

Right Angle. If one side of an angle turns a quarter of a complete circle away from the other side, the angle that is formed is a right angle (Fig. 5). It contains 90°.

When two lines intersect at right angles, the lines are per­pendicular Each angle formed by a perpendicular line con­tains 90° (Fig. 6).

Complementary angles. When two angles put together form a right angle, and thus their sum is 90°, the angles a№ complementary. For example, angle DBC is the complemen­tary of angle ABC since their sum (60°+30°) equals 90° (Fig. 7).

Straight Angle. If one side of an angle turns half a comple­te circle away from the other side, the angle that is formed is a straight angle (Fig. 8). The sides of a straight angle lie in the same straight line. Notice _A that a straight angle is twice the size of³ a right angle since in a straight angle the side has made half a complete turn,⁴ or two quarter turns. The num­ber of degrees in a straight angle is miJ

Supplementary Angles. When the sum of two angles is 180°, the angles are said to be⁵ supplementary. For example, angle ABC is the supple­mentary angle of angle CBD since their sum (120^a+60°) is 180° (Fig. 9).

Acute Angle. If one side of an angle turns less than a quarter of a circle away from the other side, the angle formed is an acute angle (Fig. 10). An acute angle, therefore, is smaller than a right angle, or less than 90°.

Obtuse Angle. If one side of an angle turns more than a quarter of a circle but less than half a circle away from the other side, the angle formed is an obtuse angle (Fig. 11). Therefore, an obtuse angle is greater than a right angle but smaller than a straight angle. It contains more than 90º but less than 180°.

Reflex Angle. If one side of an angle turns more than half

a circle (180°) but less than a complete circle (360°) away from the other side, the angle formed is a reflex angle (Fig. 12). Therefore, a reflex angle is greater than A straight angle.

Notes:

¹to depend upon — залежати від

²has turned away from — відхилений від

³twice the size of — вдвічі більше (по величині)

⁴half a complete turn — половину повного оберту

⁵angles are said to be — про кути кажуть, що вони (є)

EXERCISES

I. Read the following words paying attention to pronuncia­tion:

acute, obtuse, turn, use, unit, number, supplementary, complementary, other, but, reflex, vertex, axis, pointed, represented, straight.

II. Give words of the same root as:

Model: measure n measure v measurement v form, amount, turn, notice, determine, complement, contain.

III. Make up sentences of your own using words and expres­sions given below:

is said to be, is called, is formed, to depend upon, to turn away from, is less than, is more than, to turn more than a quarter, a degree, a circle.

IV. Answer the following questions:

1. When is an angle formed? 2. What do we call a point at which the sides of an angle meet? 3. What unit is used in measuring an angle? 4. What angles do you know? 5. How many degrees does an acute angle contain?

V. Translate into Ukrainian:

The size of measure of an angle is determined by the amount of opening between the sides, and not by the lengths of the sides.

Two angles are said to be equal if they can be placed to­gether so that their vertexes are at the same point and the two sides of one coincide with the two sides of the other. This is a very important definition.

When several lines meet at one point to form more than one angle, any two of the angles which have one side in com­mon are said to be adjacent.

When a line is drawn through the vertex of an angle be­tween the sides it is said to divide the angle.

VI. Translate into English:

Кут утворюється, коли дві прямі лінії зустрічаються в точці. Прямі лінії називаються сторонами кута, а точка, в якій вони зустрічаються — вершиною кута.

Розмір кута залежить від тієї величини (amount), на яку одна сторона відхиляється від іншої.

Градус — це одиниця виміру, що використовується при вимірюванні кута.

Кути бувають прямі, гострі, тупі. Якщо одна сторона кута відхиляється на чверть повного кола від іншої сторони, то утворений кут називається прямим кутом. Прямий кут містить 90°.

TEXT 3