Secondary Algebra

CHAPTER IV.

INTEGRAL ALGEBRAIC EQUATIONS.

We will now distinguish between two kinds of equations.

Identical Equations.

1. An example of the one kind is :

(a + b) (a - b) — a2 — b2.

The first member is reduced to the second member by performing the indicated multiplication.

2. Such an equation is called an Identical Equation, or more simply, an Identity.

3. Notice that identical equations are true for all values that may be substituted for the literal numbers involved. E.g., if a = = 5 and 6 = 3, the above equation becomes 8 × 2=25-9, or 16 = 16.

Conditional Equations.

4. An example of the second kind is:

x+1=3.

= 2.

The first member reduces to the second member, when x = It seems evident, and it is proved in School Algebra, Ch. IV., that x + 1 reduces to 3 only when x = 2.

5. Such equations impose conditions upon the values of the literal numbers involved. Thus, the equation in Art. 4 imposes the condition that if 1 be added to the value of x, the sum will be 3.

A Conditional Equation is an equation one of whose members can be reduced to the other only for certain definite values of one or more letters contained in it.

Whenever the word equation is used in subsequent work we shall understand by it a conditional equation, unless the contrary is expressly stated.

6. An Integral Algebraic Equation is an equation whose members are integral algebraic expressions in an unknown number or unknown numbers.

E.g., 3x2-4=2x, and 3x+5y= are integral equations.

7. The Degree of an integral equation is the degree of its term of highest degree in the unknown number or numbers.

8. A Linear or Simple Equation is an equation of the first degree.

E.g., x+16 is a linear equation in one unknown number.

9. A Solution of an equation is a value of the unknown number, or a set of values of the unknown numbers, which, if substituted in the equation, converts it into an identity.

E.g., 2 is a solution of the equation x + 1 = 3,

since, when substituted for x in the equation, it converts the equation into the identity 2 + 1 = 3.

The set of values 1 and 2, of x and y, respectively, is a solution of the equation x + y = 3, since 1 + 2 = 3 is an identity.

10. To Solve an equation is to find its solution.

An equation is said to be satisfied by its solution, or the solution is said to satisfy the equation, since it converts the equation into an identity.

11. When the equation contains only one unknown number, a solution is frequently called a Root of the equation.

E.g., 2 is a root of the equation x + 1 = 3.

It is evident that 8 is a root of equations (1), (2), (3), and (4). In thus applying the principles of Ch. I., Art. 17, we replace the given equation by a simpler one, which has the same root, this equation by a still simpler one, which again has the same root, and so on.

Such equations as (1), (2), (3), and (4) are called Equivalent Equations.

In general, two equations are equivalent when every solution of the first is a solution of the second, and every solution of the second is a solution of the first.

13. It is important to notice that the use of the principles. given in Ch. I., Art. 17, may lead to incorrect results.

Thus, by (iii.), we should be permitted to multiply both members of an equation by an expression which contains the unknown number.

E.g., the equation x 30 has the root 3.

Multiplying both members by x-2, we obtain

But 2 is not a root of the given equation, since 2 not equal 0.

3 does

That is, in multiplying both members by x-2, we gained a root 2. Observe that this root is the root of x -2=0. The derived equation is therefore not equivalent to the given

one.

Again, by (iii.), we should be permitted to multiply both members of an equation by 0.

Multiplying both members of x-3=0, by 0, we have

0(x-3)=0.

Any number is a root of this equation, since

0 (1 − 3) = 0, 0 (2 − 3) = 0, 0 (3 − 3) = 0, 0 (4 − 3) = 0, etc. Finally, by (iv.), we should be permitted to divide both members of any equation by an expression which contains the unknown number.

E.g., the equation (x − 1)(x + 1) = 3 (x − 1),

has the root 1, since

(1 − 1) (1 + 1) = 3(1 − 1), or 0 × 1 = 3 × 0, or 0=0;

and the root 2, since

(2-1) (2+1)=3(2-1), or 1 x 3 = 3 x 1.

Dividing both members by x − 1, we obtain

x+1=3.

This equation has the root 2 only, and not the root 1 of the given equation.

That is, in dividing both members by x 1, we lost the root 1. Observe that this root is a root of x 1=0.

The derived equation is therefore not equivalent to the given

one.

14. The correct statements of the principles which are applied in solving equations are, therefore, as follows:

(i.) Addition and Subtraction. The equation obtained by adding to, or subtracting from, both members of an equation the same number or expression is equivalent to the given one.

(ii.) Multiplication and Division. The equation obtained by multiplying or dividing both members of an equation by the same number, not 0, or by an expression which does not contain the unknown number or numbers, is equivalent to the given one.

These principles are proved in School Algebra, Ch. IV.

In the solutions of equations in the preceding chapters, we multiplied or divided only by Arabic numerals. Nevertheless, we required each result to be checked.

EXERCISES I.

Solve each of the following equations:

1. x(x+3)= x(x — 5). 2. 3x(x-5)=3x(x+2).

3. 2(x+1)-3(x+1)+ 9 (x + 1) +18=7(x+1).

4. 5(x-7) — 4(x − 7) + 11 (x − 7) = 10 + 2 (x − 7).

5. 8(3x-5)+5(3x-5)-17-2(3x-5)=