the second day and then adding the distance he went in the afternoon, or by subtracting from the first day's distance, the difference between the two distances he travelled in the morning and afternoon of the second day. Similarly, a (bc) = a b + c. We have, therefore, the general rule for 40 + 30. Removal of a Parenthesis Preceded by the Sign-. A Parenthesis, preceded by a - sign, may be removed by changing the signs of the terms that were in the parenthesis. Note. The two rules, for the removal of a parenthesis, apply equally to the removal of any of the signs of aggregation, as the bracket, brace or vinculum. In the case of the vinculum, the sign before the first of the terms under the vinculum is understood to be the sign before the vinculum. PRODUCT OF A COMPOUND BY A SIMPLE FACTOR 17. In finding the product of 3(4+2), we get the same result, whether we multiply the sum of 4 and 2 by 3, or multiply 4 by 3 and 2 by 3 and then add the products. By the first method, 3(4 + 2) = 3 × 6 = 18 X By the second method, 3(4 + 2) = 3 × 4 + 3 × 2 = 12 + 6 = 18 2) or 3(5 = 3 X 5 3 X 2 = 15 – 6 = 9 If we use algebraic symbols, as a (b + c), the second method must be used, since we cannot combine b and c by addition, as in the case of numbers like 5 and 2. We call this the distributive law for multiplication. In finding the product of any number of factors, these factors may be used in any order, without affecting the result. Perform the indicated operations and find the numerical value of each expression, when a = 2, b = 3, c = = 4, x = 5, QUOTIENT OF A COMPOUND BY A SIMPLE EXPRESSION 18. In finding the quotient of (9+6) ÷ 3, we obtain the same result, whether we divide the sum of 9 and 6 by 3 or divide 9 by 3 and 6 by 3 and then add the quotients. By the first method (96) ÷ 3 = 15 ÷ 3 = 5 By the second method (9+6) ÷ 3 = 9 ÷ 3+6 ÷ 3 = 3+2=5 Note that the value of each term should be found first and then like terms combined. EXERCISE 5 Perform the indicated operations and find the numerical value of each expression, when a = 9, b = 6, c = 3. REVIEW EXERCISE II 2. When a = 1, b = 2, c = 3, d = 4, e = 5, find the value of 6. Find the value of x3+ y3 + z3 — 3xyz, if x = 1, my2 — (y — x) + axy2 + (x x) + axy2 + (x − y)2 + d3. value of xyz2. = 2, z = 1, c = 0, find the value of - (x − y) (y — 2) — y3z + (y + c) — zo + 9. If a 7a2bc (2a - b) + (a — c)2 + 14abcd + ac+1. CHAPTER III SIMPLE EQUATIONS 19. Equation. An equation is a statement of equality between two algebraic expressions. = Thus, if we wish to state that the expression 2x + 3 is equal in value to the expression 3x + 2, we write this statement as an equation, 3x + 2 2x + 3. This means that the two numbers, represented by the two expressions, 3x + 2 and 2x + 3, are equal numbers or the same number. The two equal expressions are called the members or sides of the equation. The one, to the left of the sign of equality, is the left member and the other, the right member of the equation. If the statement of equality is true for all values of the letters used, the equation is called an identical equation or simply an identity. Thus, the equation, a + b = b + a is an identity since. the two members of the equation will have equal numerical values for any values of a and b. It is also evident, that any equation that does not contain letters will be an identity, as, 5 + 3 4 X 2. = If we consider the equation, 3x + 2 = 2x + 3, the statement may be read, three times the number x plus two must equal twice the number x plus three. In other words, the value of x must be such as to agree with the condition expressed by this equation. It can easily be seen that the two members will only be equal if x is equal to 1. Such an equation is therefore called an equation of condition or simply an equation. Since this value of x satisfies the condition expressed by the equation, we say that the equation is satisfied by this value of x. Equations are usually employed in finding the value of |