Literal Equations and Formulas


 Vanessa Dorothy McCormick
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1 . Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express a relatioship betwee more tha oe variable or letter. You are already familiar with a umber of examples. For istace, P R B A 1 h b P L W NOTE A literal equatio is ay equatio that ivolves more tha oe variable or letter. are formulas for percetage, the area of a triagle, ad the perimeter of a rectagle, respectively. Oe useful applicatio of the equatiosolvig skills we cosidered i Sectio.1 is i rewritig these formulas, also called literal equatios, i more coveiet equivalet forms. Geerally, that more coveiet form is oe i which the origial formula or equatio is solved for a particular variable or letter. This is called solvig the equatio for a variable, ad the steps used i the process are very similar to those you saw earlier i solvig liear equatios. Cosider the followig example. Example 1 Solvig a Literal Equatio Solve the formula d r t for t This formula gives distace d i terms of a rate r ad time t. To solve for t meas to isolate t o oe side of the equatio. This ca be doe by dividig both sides by r. Give d r t 001 McGrawHill Compaies we use the multiplicatio property of equatios to divide by r, the coefficiet of t. d r t r r d r t We usually write the equatio i the equivalet form with the desired variable o the left. So t d r We ow have t i terms of d ad r, as required. CHECK YOURSELF 1 Solve the formula C pr for r. 63
2 64 CHAPTER LINEAR EQUATIONS AND INEQUALITIES Solvig a formula for a particular variable may require the use of both properties of equatios, as the followig example illustrates. Example Solvig a Literal Equatio Solve the formula P L W for L NOTE We wat to isolate the term with the variable we are solvig for here L. This formula gives the perimeter of a rectagle P i terms of its width W ad its legth L. To solve for L, start by usig the additio property of equatios to subtract W from both sides. P L W P W L W W P W L We ow use the multiplicatio property to divide both sides by : NOTE This result ca also be writte as L P W P W P W L L L P W This gives L i terms of P ad W, as desired. CHECK YOURSELF Solve the formula ax by c for y. You may also have to apply the distributive property i solvig for a variable. Cosider the followig example. Example 3 Solvig a Literal Equatio Solve the formula A P(1 rt) for r This formula gives the amout A i a accout earig simple iterest, with pricipal P, iterest rate r, ad time t. First, we use the distributive property to remove the paretheses o the right. A P(1 rt) P Prt 001 McGrawHill Compaies
3 LITERAL EQUATIONS AND FORMULAS SECTION. 65 We ow subtract P from both sides. A P P P Prt A P Prt Fially, to isolate r, we divide by, the coefficiets of r o the right. A P A P Prt r r A P CHECK YOURSELF 3 Solve the equatio for. S 180( ) Ofte it is ecessary to apply the multiplicatio property, to clear the literal equatio of fractios, as the first step of the solutio process. This is illustrated i Example 4. Example 4 Solvig a Literal Equatio Solve the formula for C. D C S This formula gives the yearly depreciatio D for a item i terms of its cost C, its salvage value S, ad the umber of years. As our first step, we multiply both sides of the give equatio by to clear of fractios. 001 McGrawHill Compaies NOTE O the right ote that 1 ad multiplyig by 1 leaves C S. D C S C S D D C S We ow add S to both sides. D S C S S D S C C D S ad the cost C is ow represeted i terms of, D, ad S.
4 66 CHAPTER LINEAR EQUATIONS AND INEQUALITIES CHECK YOURSELF 4 Solve the formula V 1 3 pr h for h. CHECK YOURSELF ANSWERS c ax S r C. y h 3V p b 180 pr 001 McGrawHill Compaies
5 Name. Exercises Sectio Date I exercises 1 to 4, solve each of the formulas for the idicated variable. 1. V Bh for h. P RB for B ANSWERS C pr for r 4. e mc for m V LWH for H 6. I Prt for r V pr h for h 8. S prh for r V Bh for B 10. V pr h for h 3 3 E KT 11. I for R 1. V for T R P ax b 0 for x 14. y mx b for x P L W for W 16. ax by c for y C S R(100 x) 17. D for S 18. D for R R C(1 r) for r 0. A P(1 rt) for t McGrawHill Compaies 1 1. A h (B b) for b. L a ( 1)d for F C 3 for C 4. C (F 3) for F
6 ANSWERS Solve each of the followig exercises usig the idicated formula from exercises 1 to A rectagular solid has a base with legth 6 cm ad width 4 cm. If the volume of the solid is 7 cubic cetimeters (cm 3 ), fid the height of the solid. See exercise A cylider has a radius of 4 iches (i.). If its volume is 144p cubic iches (i. 3 ), what is the height of the cylider? See exercise A pricipal of $000 was ivested i a savigs accout for 4 years. If the iterest eared for that period was $480, what was the iterest rate? See exercise The retail sellig price of a item, R, was $0.70. If its cost, C, to the store was $18, what was the markup rate, r? See exercise The radius of the base of a coe is 3 cm. If the volume of the coe is 4p cm 3, fid the height of the coe. See exercise The volume of a pyramid is 30 i. 3. If the height of the pyramid is 6 i., fid the area of its base, B. See exercise 9. 6 i. 31. If the perimeter of a rectagle is 60 ft ad its legth is 18ft, fid its width. See exercise The yearly depreciatio, D, for a piece of machiery was $1500 over 8 years. If the cost of the machiery was $15,000, what was its salvage value, S? See exercise A pricipal of $5000 was ivested i a timedeposit accout payig 9% aual iterest. If the amout i the accout at the ed of a certai period was $750, for how log was the moey ivested? See exercise The area of a trapezoid is 36 i.. If its height is 4 i. ad the legth of oe of the bases is 11 i., fid the legth of the other base. See exercise 1. Aswers 1. h V 3. r C 5. H V 7. h V 9. B 3V B p LW pr h P L 11. R E 13. x b 15. W 17. S C D I a A hb C 5 5F 160 r R C b (F 3) or C C h cm 7. 6% 9. 8 cm ft years 001 McGrawHill Compaies 68
.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
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