some unknown number from its relation to certain known numbers. This relation is expressed algebraically by the equation. Unknown numbers are usually represented by the letters at the end of the alphabet, as x, y, z, and the known numbers either by the Arabic numerals or the letters at the beginning of the alphabet, as a, b, c.

20. Simple Equation.-When an equation is reduced to its simplest form and then contains only the first power of the unknown numbers, it is called a simple equation or equation of the first degree. Thus, 2x + 3 = 9 and ax b c are simple equations in x.

Solution of Equations. Finding that value of the unknown number, which will satisfy the condition expressed by the equation, is called solving the equation.

21. Roots of the Equation. The value of the unknown number, which satisfies the equation, is called a root of the equation and is usually found by solving the equation. It is important for the

student to remember that the two sides of the equation always represent equal numbers and that this equality must never be disturbed. An equation may

2x +3

たんぱ

be likened to a balance, having equal weights on the two pans of the balance. To keep the two sides balanced, they must be treated in the same way; whatever is done to one side must also be done to the other. If a weight is added to one side, an equal weight must be added to the other to maintain a balance. Similarly, if a weight is removed from one side, an equal weight must be removed from the other side.

In solving an equation, we therefore observe certain rules based upon truths called axioms, statements so selfevident that they are accepted as true.

Ax. 1. If equal numbers are added to equal numbers, the sums are equal.

Ax. 2. If equal numbers are subtracted from equal numbers, the remainders are equal.

Ax. 3. If equal numbers are multiplied by equal numbers, the products are equal.

Ax. 4. If equal numbers are divided by equal numbers, the quotients are equal.

Ax. 5. If two numbers are equal to the same number, they are equal to each other.

22. Transposition of Terms. In solving equations, it becomes necessary to transfer terms from one side of the equation to the other. This is called transposing the terms. 1. Solve the equation, 3x + 2

= 11

To find the value of x, which will satisfy this equation, the term containing x should be kept by itself on one side of the equation. Therefore we must transpose the term 2 to the other side. According to axiom 2, we may subtract equal numbers from each side of the equation. Subtracting 2 from each side, we have,

Therefore

3x + 2

- 2 = 11 2
3x = 9

By Axiom 4, we may divide each side of the equation by the same number.

Dividing each side by 3, we have, x = 3.

2. Solve the equation, x b:

= α

As in the preceding problem, we must transpose the term -b to the other side of the equation. By Axiom 1, equal numbers may be added to each side of the equation.

In the first problem, we transposed the term 2 from the left to the right side, by subtracting 2 from each side. The effect would have been the same if we had transferred the 2 to the right side of the equation and changed its sign to minus. Similarly, in Problem 2, instead of adding b to each side, we could have transposed the term -b to the right side of the equation by changing its sign to plus.

Hence we have the general Rule for Transposition:

Any term may be transposed from one side of the equation to the other, by changing its sign.

The effect is the same if we simply cancel the equation.

4. Solve the equation, y + a = c + a. Transposing a to the right,

y = c + a — a

+a and -a, cancel, hence

c on each side of the

y = c

We see from Problems 3 and 4, that

Equal terms occurring on both sides of an equation may be cancelled.

If we transpose every term of the equation

The sign of every term of an equation may be changed without disturbing the equality.

23. Numerical Equations.-An equation, in which all the known numbers are expressed in Arabic numerals, is called a numerical equation.

Transposing -5 to the right side and -3x to the left side,

Transposing all terms containing the unknown number x to the left and the known numbers to the right,

From the above solutions, we have the general Rule for the Solution of a Simple Numerical Equation.

Simplify the equation, transpose all the unknown terms to the left, the known terms to the right. Combine like terms and finally divide both sides of the equation by the coefficient of the unknown number (x or any other letter representing the unknown number.)

Note. The word simplify in Algebra should be interpreted as perform the indicated operations. In the solution of the two problems above, it will be seen that the original forms of the equations contained indicated operations of addition, subtraction and multiplication. These operations were performed before the terms could be transposed.

VERIFICATION

24. If, when an equation has been solved, the value of the unknown is substituted in the original equation and this equation is then reduced to an identity, the root of the equation is said to be verified.

Thus in problem (1, we found x = 3.

Substituting this value for x in the original equation

15. 5(x-3) 7(6x) + 29 563(10 − x).

=

STATEMENT OF PROBLEMS

25. The one great object of Algebra is to simplify the solution of problems. It is very important that the student should learn to translate the statements of a problem into algebraic language. The following exercise is designed to give the student facility of translation before considering the actual solution of problems.

EXERCISE 7

1. Express in algebraic symbols: x increased by y; r diminished by s; a multiplied by b; v divided by t; the square of z; the cube root of v; one half of a multiplied by the square of t.

2. If one part of 30 is 12, what is the other part? 3. If one part of 25 is x, what is the other part? 4. If one part of x is a, what is the other part?

5. If the sum of two numbers is 25, and one of them is 10 what is the other?

6. If the sum of two numbers is 40 and one of them x, what is the other?

7. If the sum of two numbers is s and one of them is a, what is the other?

8. If the difference between two numbers is 8, and the smaller number is 10, what is the larger number?

9. If the difference between two numbers is d, and the smaller number is a, what is the larger number?

10. If the difference of two numbers is 5, and the larger number is x, what is the smaller number?

11. By how much does 20 exceed 15?

12. What is the excess of 7x over 3x?