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46. Changing Signs throughout an Equation. The signs of all the terms of an equation may be changed without destroying the equality. Thus, 6–4 x=10–3 x is equivalent to –6+4 x= -10+3 x. In fact, all we have to do to get this last equation is to multiply both sides of the first equation by –1. (Axiom III, § 9.)
Changing signs throughout an equation in this way is often useful, as illustrated below : EXAMPLE. Solve the equation
–4 x+3= -x-9. SOLUTION. Changing signs throughout gives 4 x3=x+9. Transposition gives
3 x=12, and hence
47. Definitions. Any equation which, like those we have thus far considered, contains no higher power of the unknown letter than the first is called a simple equation, or an equation of the first degree, or a linear equation.
Thus, 3 x-4=4 x–5 is a simple equation, but 3 x2–4=4 x–5 is not a simple equation. Other examples of equations that are not simple are: x3+2 x2=x+4; 3 x3—4=x+5; x++3 x2— x=7.
Equations containing no higher power of the unknown letter than the second will be considered in later chapters.
The value of the unknown letter in a simple equation is called the root, or the solution, of the equation. To solve a simple equation is (as we have seen) to find its root.
The usual process of solving consists in making full use of the short methods explained in this chapter, such as transposition, cancellation of terms, etc., so as to arrive quickly at the value of the unknown letter.
Solve each of the following equations, making use of the short methods explained in this chapter in any way you please. Check your answer.
1. x +2=2 x-1. [HINT. Transpose so as to have all x terms on the left and all other terms on the right. Then combine like terms and use Axiom IV.) 2. 2x+1=-2-3.
4. 1 *—3=x+5.
5. x+1= x+1.
10. If 1 be added to twice a certain number, the result is 3 more than the number itself. What is the number?
[Hint. Work by algebra, letting x represent the unknown number.]
11. If a certain number be subtracted from three times itself, the result is 5 more than the number. What is the number?
12. Answer the last question when – 5 is used instead of 5. What does the question mean in this case ?
13. A certain recipe for fruit punch calls for twice as many oranges as lemons. If oranges are 35 cents a dozen and lemons are 25 cents a dozen, how many dozens of each must be used in making a punch that is to cost $4.75?
[Hint. Let x=the number of dozens of lemons used.]
14. In a certain house the parlor contains 3 more lights than the dining room, and the dining room 4 more than the kitchen, while the pantry contains one light only. If the total is 24 lights, how many are there in each of the rooms?
15. The interest on a certain sum of money at 5% amounts in one year to $1 less than it would at 6%. What is the sum?
16. The month of March contained 13 more stormy days than bright ones, and of the stormy days there were 3 more with snow than rain. How many were there of each kind ?
17. In trying to find the weight of a single egg, I found that four eggs and one ounce weight balanced against one egg and a half pound weight. What is the weight of each egg?
For further exercises on this topic, see Appendix, p. 297. ,
MULTIPLICATION AND SPECIAL CASES OF FACTORING
PART I. MULTIPLICATION 48. Product of Powers of the Same Number. We have seen (§ 13) that x2 (read x square, or x to the second power) means x · X, while x3 (read x cube, or x to the third power) means x · X · X.
Likewise, we explained ($ 25) that x4 (read x to the fourth power) means x · X · X · X.
In the same way x5 (read x to the fifth power) means X.: X · X · X ·x, etc.
In each case the number above the x is called the exponent of x.
Suppose now we consider the product x2 • 23. This means (x • 2) - (x · X · x). But this is the same as x · X · X · X ·X, which is x5. Thus, we see that
x2 • X3 = x(2+3) = 25. Similarly, if we consider x3 • 24, we may write
23 • 24 = x · X · X · X ·x · X · x = x(3+4) = x7. We arrive in this way at the following rule. RULE FOR MULTIPLYING POWERS OF THE SAME NUMBER. The exponent of the product of two powers of the same number is equal to the sum of the exponents of the factors.
Stated as a formula, this rule becomes
(Leonhard Euler, 1707-1783) Revised and enlarged all the branches of mathematics known at his time. In algebra, especially famous for his studies on Infinite Series. His great work entitled Analysis Infinitorum contains essentially all that is to be found to-day in the textbooks on algebra and trigonometry, besides much else of a more advanced character.