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Here we proceed as follows. Subtracting 3 x from both sides, 6+3 x=12.

(Axiom II) Subtracting 6 from both sides, 3 x=6.

(Axiom II) Dividing both sides by 3,

x=2. Ans. (Axiom IV) Thus, the axioms in § 9 enable us to solve all such equations. We may sum up our results in the following rule :

RULE FOR SOLVING EQUATIONS. Use axioms I and II to get rid of all the terms, or parts, on one side of the equation that contain the letter, and to get rid of all terms, or parts, on the other side of the equation that do not contain the letter. Then use axioms III and IV to find the value of the letter.

ORAL EXERCISES In each of the following equations, tell what to do to get rid of the term on the right side that contains the letter. 1. 3=5– x.

7. 3 y-1=y+2. 2. 4=20—2 x.

8. 82-1=3–2. 3. 5 x=3 x+10.

9. gs+4=;s +8. 4. 6r-2=2r+10.

10. 21 a-5=į a-2. 5. 10+5 c=30-15 c. 11. .5 x= 2 x+.3 6. 5 m+2=6+m.

12. .5x={x—.08 In each of the following equations, tell what must be done to the equation to get rid of the term on the left side that contains no letter: 13. 4x+3=15. 16. 10+6 z=3 z+40. 19. 5 m-1=2 m+i 14. 38–8=16. 17. 4x-1=2 x+6. 20. Ar-8=fr+3. 15. 5+4 a=21. 18. įrt}={r+. 21. 6 m-23=m+2.

WRITTEN EXERCISES. Solve each of the following equations and check your answer in each case. 1. 5x – 2=x+10.

7. 20+7 x–5 x=48-4 x.. 2. 10 r+4=6r+16.

8. 6 x– 15–9=4 x–2 . 3. 4+5 a=11-2 a.

9. 2 x+3 x+5-x=8+x. 4. 13 y-99=7 y+69. 10. į x+2 x-3=x+1. 5. 62–4=2z+20.

11. 4x+1=1 x+2. 6. 12 x – 2 x=20+5 x. 12. .3 x+5=4x+1.3

13. Find the number which added to three times itself gives 36. SOLUTION.

Let x=the number sought. Then,

3x+x=36, hence,

4x=36, and therefore,

x=9. Ans. 14. If 8 is subtracted from 5 times a certain number, the result is equal to the number increased by 2. What is the number? (See Ex. 13.)

15. The water and steam in a boiler occupied 120 cu. ft., and the water occupied twice as much space as the steam. How many cubic feet did each occupy?

[HINT. After finding x, the space occupied by the steam, it is necessary to multiply it by 2 to find the space occupied by the water.]

16. In a fire, B lost twice as much as A, while C lost three times as much as A. If their combined loss was $6000, what was the loss of each?

17. A, B, and C begin business with a capital of $7500. A furnishes twice as much as B, while C furnishes $1500. How much does A furnish, and how much does B furnish?

18. A boy bought a bat, a ball, and a glove for $2.25. If the bat cost twice as much as the ball, and the glove three times as much as the bat, what was the cost of each?

19. The sum of the angles of any triangle is 180°. In the triangle ABC the angle at A is three times as large as the angle at B, while the

Fig. 6. angle at C is twice that at A. What is the number of degrees in each?

20. A certain rectangle is four times as long as it is wide. The distance around it is 200 rods. Find its dimensions (length and breadth).

(HINT. The opposite sides of a rectangle are equal.]

13. Squares and Cubes. The product of two equal numbers is called the square of that number. Thus, X • x=x?, and is read “x square,” just as in arithmetic 3X3=32 is read “ three square.”

The product of three equal numbers is called the cube of that number. For example, X • x • x=x3, and is read “ X cube.”

14. Square Root and Cube Root. If there are two equal factors of a number, either is called the square root of the number. Thus, 25 has two equal factors, 5 and 5; hence the square root of 25 is 5. This is written V 25=5. In the same way, V x2 = x.

Similarly, if there are three equal factors of a number, either of them is called the cube root of that number. Thus, 64 has three equal factors 4, 4, and 4; hence the cube root of 64 is 4. This is written V64=4. In the same way,

x3 = x.

ORAL EXERCISES Read each of the following expressions. 1. x2+x3. 4. a?ta3+ Va. 7. a Vx+b Vy. 2. x– Væ. 5. Vm+ m+c?. 8. gx2+x+i. 3. y3+x. 6. 40z+2V2. 9. pvy-qor+1.

15. Order of Operations. Operations are performed in the following order :

First, all multiplications and divisions in their order from left to right.

Second, all additions and subtractions in their order from left to right.

Thus, 6+8.3, means six plus the product of eight and three; that is, 6+8.3=6+24=30.

EXAMPLE 1. Find the value of 3.2+4.3-+22.
SOLUTION. 3. 2+4.3—+22=6+12–3+4=19. Ans.
EXAMPLE 2. If x=2, y=3, 2=4, find the value of

5 xy+6 22 –
SOLUTION. Giving x, y, and a their values, we have
5.2.3+6.42 - 54 = 5.2.3+6.16 – 4

= 30+ 96 -4 = 122. Ans.

ORAL EXERCISES State the value of each of the following expressions. 1. 6+8.2-7.

6. 62.2-6.3+kk 2. 10–3.2+1,2

7. 22.32 – 3:22 3. 2-4+10.

8. 10+2+3-40 4. 20—3.4+6+3+6.

9. 3.42—4.2+1. 6.5 10

10. 23+3.4.2-3.5.2

3

-5 +8.

11. V25+9+2V16

12.
12.

8 319 2-1+ 27

WRITTEN EXERCISES

10. 3b-4a+10 c

13. 1+*+

If a=1, b=2, c=3, and d=4, find the value of each of the following expressions. 1. 562c. 2. 4c+3 a. 3. d-26. 4. c+2 d. 6. d+b-2 c. 6. 30-3 a. 7. abcd 3 bc. 8. 2b+d_2a.

- 2 a [Hint. In solving Ex. 8, remember that the numerator, 26+d, must be found first, then divided by 2 a.) 9. 26+c-d.

"+b. с

2d 4c+5 b-a

b+2 c+4 bc
.. dtc

6+2 d+3 62
ab , bc, cd

C a' b
15. 6 d2—4 c+5 62+1. 16. a?+62+c+d2.
17. 4 ab2+6 bc2 — 2 cda. 18. 8ab+ vā-de.

Va+b2+ao 2 a+3 6+c

C2 - 2 21. Vbd+ Vad.

abd+a2b2c3

b+Vă– 3 23. aụb V4a+608 a.

- c2_b 24. Write in words the meaning of the expressions in exercises 1 to 8. For example, Ex. 1 means “five times b minus two times c.".

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