8. (73 x2-25 +56 +95 x 59 x3) 59203) (11x+7x3-3x2+1). 9. (49 x3-72 xy2 + 28 y3) ÷ (7 x -3y). 10. (4 m2 m2n2 + 6 mn3 — 9 n3) ÷ (2 m2 — mn + 3 n2). 12. For what value of k is 2-3x+k exactly divisible by x+1? The division is exact if the remainder is 0, or if k + 4 = 0, that is, if k = 4. 13. Determine k so that 3 + 3 x2 + 2 x + k shall be exactly divisible by x − 2. 14. Determine k so that x+1 shall be an exact divisor of 23+k. 15. For what value of k is x − 1 an exact divisor of 3 + k? VIL. SIMPLE EQUATIONS 162. An equation is a statement expressing the equality of two numbers. (Review §§ 12–16.) There are two essentially different kinds of algebraic equations as illustrated by the following: Equation 1 is true for all values of a; equation 2 is satisfied when x equals 3, and not otherwise. 163. Identity. An equation that is true for all values of the letters involved is an identical equation, or simply an identity. The symbol is sometimes used to indicate an identity. The most frequent use of the identical equation is to indicate the result of some operation performed upon algebraic expressions. The following are examples of identical equations: In the identical equation, if the indicated operations are performed and the like terms are collected in each member, the two members will be exactly alike. 164. Conditional Equation. An equation that is true for only certain values of the letters involved is a conditional equation or simply an equation. A conditional equation expresses a relation between an unknown number and certain known numbers. The problem suggested by a conditional equation is that of finding for what value of the unknown number the relation expressed in the equation is true. The following are examples of conditional equations: 2x-7=x+3. 3 a +7=4a +7. True when x = 10, and not otherwise. True when a = O and not otherwise. ORAL EXERCISE 165. 1. Is x+1=2 a conditional equation or an identity? 2x+3=7? 2. Is 2x-(x+1)=x-1 a conditional equation or an identity? (x-1)(x+1)= x2-1? 2x-1=x? 3. State the four principles used in solving equations. (See § 13.) 4. What is the root of an equation? (See § 16.) 5. What is the root of x+2=7? of x-2=7? of 2x=3? of x=5? 6. What value of x satisfies the equation x-2=3? Show that the following are identities by reducing the two members to the same expression : EXERCISE 166. Show that equations 1 to 4 are identical equations by reducing the two members to the same expression. 1. a(bc)+b(ac)= 2 ab - c(a + b). 2. (a+b-c)(a+b+c) = a2 +2ab+b2 — c2. 3. (x2-x-2) (x2 + x − 2) = (x2 - 3x+2) (x2+3x+2). 4. (x + y)(y + z) (z + x) + xyz = (x + y + z)(xy + yz + zx). By substituting 1, 2, and 3 for x in equations 5 to 8, show that each is a conditional equation. 5. 2x-5=x — 3. 7. (x-1)(x+2)= x2. 6. (x-4)2+2=(x-5)2- 3. 8. 8x+7x=14. 14. 7y-9-3y+5=11y-2(3+2y). 15.405-30x+35-40 x. 16. bb9 +36 +2. 167. It is a common practice in algebra to use x, y, and z to represent the unknown numbers in an equation and to use a, b, c etc. to represent numbers that are regarded as known. Thus, in the equation ax = 3 a2b, x is the unknown number and the value of x is to be found in terms of the other letters involved. The value of x is found by dividing both members of the equation by a, giving x = 3 ab. The equation x 3 ab can be solved for a or for b. Thus, dividing both members by 3b gives a = Solve the equation for b. 168. Integral Equation. An equation in which the unknown number does not occur in any denominator is an integral equation. Let the student check the answer by putting 4 for x in the original equation. = 4 x 16 is the simplified form of the equation and the work done to reduce the equation to this form is called simplifying the equation. 2. Solve (x-3)(x − 2) = (x − 4)2. Let the student check the answer as in example 1. What is the simplified form of this equation? 170. Simple Equation. An equation that can be reduced to an integral form containing the first power of the unknown number and no higher power is a simple equation. Thus, 5 x − 2(3 x − 1) = 1 is a simple equation. Also x(x − 5) = (x − 3) (x7) is a simple equation, for it reduces to 5 x = 21. Simple equations are frequently called first degree equations, also linear equations. |