Elementary Algebra

25. Find the product of (2a+3b) (2a+3c−2b), and test the result by making a = 1, b=4, c=2.

26. If a, b, c, d, e..... denote 9, 7, 5, 3, 1, find the values of

(a−b−c)2+(b−a−c)2 + (c− a−b)2 when a == 1, b=2, c = 3.

30. Find the value of

(a+b−c)2 + (a−b+c)2 + (b+c− a)2 when a=1, b=2, c=4. 31. Find the value of

(a+b)2 + (b+c)2+(c+a)2 when a= −1, b=2, c=-3.

32. Shew that if the sum of any two numbers divide the difference of their squares, the quotient is equal to the difference of the two numbers.

33. Shew that the product of the sum and difference of any two numbers is equal to the difference of their squares.

34. Shew that the square of the sum of any two consecutive integers is always greater by one than four times their product.

35. Shew that the square of the sum of any two consecutive even whole numbers is four times the square of the odd number between them.

36. If the number 2 be divided into any two parts, the difference of their squares will always be equal to twice the difference of the parts.

37. If the number 50 be divided into any two parts, the difference of their squares will always be equal to 50 times the difference of the parts.

38. If a number n be divided into any two parts, the difference of their squares will always be equal to n times the difference of the parts.

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39. If two numbers differ by a unit, their product together with the sum of their squares is equal to the difference of the cubes of the numbers. яну жу

40. Shew that the sum of the cubes of any three consecutive whole numbers is divisible by three times the middle number.

VI. ON SIMPLE EQUATIONS.

106. AN EQUATION is a statement that two expressions are equal.

107. An Identical Equation is a statement that two expressions are equal for all numerical values that can be given to the letters involved in them, provided that the same value be given to the same letter in every part of the equation.

Thus,

(x+a)2= x2+2ax+a2

is an Identical Equation.

108. An Equation of Condition is a statement that two expressions are equal for some particular numerical value or values that can be given to the letters involved.

is an Equation of Condition, the only number which x can represent consistently with this equation being 5.

It is of such equations that we have to treat.

109. The Roor of an Equation is that number which, when put in the place of the unknown quantity, makes both sides of the equation identical.

110. The Solution of an Equation is the process of finding what number an unknown letter must stand for that the equation may be true: in other words, it is the method of finding the Root.

The letters that stand for unknown numbers are usually x, y, z, but the student must observe that any letter may stand for an unknown number.

111. A SIMPLE EQUATION is one which contains the first power only of an unknown quantity. This is also called an Equation of the First Degree.

112. The following Axioms form the groundwork of the solution of all equations.

Ax. I. If equal quantities be added to equal quantities, the sums will be equal.

Ax. II. If equal quantities be taken from equal quantities, the remainders will be equal.

Ax. III. If equal quantities be multiplied by equal quantities, the products will be equal.

Ax. IV. If equal quantities be divided by equal quantities, the quotients will be equal.

113. On Axioms I. and II. is founded a process of great utility in the solution of equations, called THE TRANSPOSITION OF TERMS from one side of the equation to the other, which may be thus stated.

"Any term of an equation may be transferred from one side of the equation to the other if its sign be changed."

Then, by Ax. I., if we add a to both sides, the sides remain equal:

Then, by Ax. II., if we subtract c from each side, the sides remain equal :

114. We may change all the signs of each side of an equation without altering the equality.

115. We may change the position of the two sides of the equation, leaving the signs unchanged.

Thus the equation a−b=x-c, may be written thus,

x-c-a-b.

116. We may now proceed to our first rule for the solution of a Simple Equation.

RULE I. Transpose the known terms to the right hand side of the equation and the unknown terms to the other, and combine all the terms on each side as far as possible.

Then divide both sides of the equation by the coefficient of the unknown quantity.

This rule we shall now illustrate by examples, in which stands for the unknown quantity.

Ex. 1. To solve the equation,