Exercises

I. Answer the following questions on the text:

1. What is an algebraic expression?

2. What operations algebraic expressions should be performed first?

, ,ill!!

3. What are the operations of the higher order called?

4. What does A stand for in the formula A = bl?

5. What does b stand for?

6. What does l stand for?

II. Give the Hungarian for the following words and word combinations. Use them in sentences of your own:

order of operations, the numerical value of the expression, to perform the operations, involution, devolution, the area of a rectangle, to square a number.

Lesson 7 equations and proportions

Equations. The idea of an equation is very simple because equation only shows that two quantities are equal. The statement «Three times a certain number added to five is equal to twenty six» is an equation. The same statement can be written in the language of algebra in such a way:

3n + 5 = 26 in which the letter n stands for the unknown number.

When the unknown number is found, we say that «the equation has been solved».

The numbers represented by letters are called the “unknowns” (or “unknown numbers”). These numbers are usually represented by the last few letters of the Latin alphabet (...x, y, z). An equation may have one, two or more unknowns.

An equation does not change if:

1) the same expression is added to or subtracted from both numbers of the equation;

2) both members are multiplied or divided by the same known number, except zero.

The values of the unknowns that satisfy the equation are called the root of the equation. An equation in one unknown may have one, two or more roots. For example, 5x–7 = 8 has only one root (3), the equation y² + 2 = 3y has two roots (1 and 2), and the equation (x–2) × (x–1) = 0 has three roots (I, 2 and -I).

Solution of some problems:

1) The difference between two numbers is 24, and the greater number is 4 times the less number. What are the numbers?

To solve this problem we take the letter x for the less number. Then 4x = the greater number. Their difference is 4x–x. Hence, 4x–x = 3x; 3x =24, x = 8. That is the less number; the greater number is 4x, or 32.

2) The sum of two numbers is 30, and the greater exceeds the less by 4. What are the numbers?

We may represent the less number by x. Then greater number will be represented by x + 4. By the condition of the problem, the sum of the less number and the greater is 30. We may write it down in the algebraic language as follows:

x + (x + 4) = 30.

Now we collect similar terms: 2x + 4 = 30. The members of this equation are equal numbers. If from each of them we subtract the number 4, the resulting number will be equal:

2x = 30 - 4

2x = 26

x = 13

Hence the less number is 13 and the greater is 13 + 4 = 17.

Proportions. We know that the relation between two numbers or two amounts expressed by dividing one by the other is called ratio. For example, the ratio of 60 : 5 is 12.

An equality of ratios is called a proportion. 12 : 3 =20: 5 and c : x = d : y is examples of proportions. The proportion c : x = d : y is read “the ratio of c to x is equal to the ratio of d to y”. It may be also read in another way: “c to x as d is to y”.

In a proportion, the first and the fourth terms are called the extremes, and the second and the third terms are called the means. In a : b = c : d; a and d are the extremes, b and c are the means.

In any proportion the product of the extremes is equal to the product of the means.