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 — уравнения с двумя не известными

² an alternative way — другой способ

³ so that — так что; таким образом, что

⁴ in order to find — для того, чтобы найти

Exercises

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

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

2. 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.

3. 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 Russian:

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.

4. Translate into English:

Уравнением называется равенство, в котором одно или несколько чисел, обозначенных буквами, являются неизвестными.

Пусть, например, сказано, что сумма квадратов двух неизвестных чисел х и у равна 7; это можно записать при помощи следующего уравнения с двумя неизвестными

x² + у² = 7.

Уравнением первой степени с двумя неизвестными называется уравнение вида

ах + bу = с,

где х и у — неизвестные, а и b (коэффициенты при неизвестных) — данные числа, не равные оба нулю, с (свободный член — absolute term) — любое данное число.

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