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EXAMPLE 1

= (–5) (–5) (–5) (–5). (–5) 4. = –(5 5 5 5). –5 4. EXAMPLE 1. Evaluate powers. = 625. = –625. EXAMPLE 2. Evaluate an algebraic expression. Evaluate –4 x 2 –6 x + 11 when x = –3. = – 4( –3 ) 2 – 6( –3 ) + 11. –4 x 2 –6 x + 11. Substitute – 3 for x.

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EXAMPLE 1

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  1. = (–5) (–5) (–5) (–5) (–5)4 = –(5 5 5 5) –54 EXAMPLE 1 Evaluate powers = 625 = –625

  2. EXAMPLE 2 Evaluate an algebraic expression Evaluate –4x2–6x+ 11 when x = –3. = –4(–3)2–6(–3) + 11 –4x2–6x+ 11 Substitute –3 for x. = –4(9) –6(–3) + 11 Evaluate power. = –36 + 18 + 11 Multiply. = –7 Add.

  3. Write an expression that shows your profit from selling ccandles. EXAMPLE 3 Use a verbal model to solve a problem Craft Fair You are selling homemade candles at a craft fair for $3 each. You spend $120 to rent the booth and buy materials for the candles. • Find your profit if you sell 75 candles.

  4. – – 3 c 120 EXAMPLE 3 Use a verbal model to solve a problem SOLUTION STEP1 Write: a verbal model. Then write an algebraic expression. Use the fact that profit is the difference between income and expenses. An expression that shows your profit is 3c –120.

  5. Your profit is $105. ANSWER EXAMPLE 3 Use a verbal model to solve a problem STEP2 Evaluate: the expression in Step 1 when c = 75. 3c –120 = 3(75) – 120 Substitute 75 for c. = 225 – 120 Multiply. = 105 Subtract.

  6. 8x + 3x 5p2 + p – 2p2 3(y + 2) – 4(y – 7) EXAMPLE 4 Simplify by combining like terms = (8 + 3)x Distributive property = 11x Add coefficients. = (5p2– 2p2) + p Group like terms. = 3p2 + p Combine like terms. = 3y + 6 – 4y + 28 Distributive property = (3y – 4y) + (6 + 28) Group like terms. = –y + 34 Combine like terms.

  7. 2x – 3y – 9x + y EXAMPLE 4 Simplify by combining like terms = (2x – 9x) + (– 3y + y) Group like terms. = –7x – 2y Combine like terms.

  8. Digital Photo Printing EXAMPLE 5 Simplify a mathematical model You send 15 digital images to a printing service that charges $.80 per print in large format and $.20 per print in small format. Write and simplify an expression that represents the total cost if nof the 15 prints are in large format. Then find the total cost if 5 of the 15 prints are in large format.

  9. EXAMPLE 5 Simplify a mathematical model SOLUTION Write a verbal model. Then write an algebraic expression. An expression for the total cost is 0.8n + 0.2(15 – n). 0.8n + 0.2(15 – n) = 0.8n + 3 – 0.2n Distributive property. = (0.8n – 0.2n) + 3 Group like terms.

  10. ANSWER When n = 5, the total cost is 0.6(5) + 3 = 3 + 3 =$6. EXAMPLE 5 Simplify a mathematical model = 0.6n + 3 Combine like terms.

  11. Solve x + 8 = 20. x + 8 = 20 x = (12) x = 12 4 5 Multiply each side by , the reciprocal of . 4 5 The solution is 15. ANSWER CHECK x = 15 in the original equation. 4 4 4 4 5 4 5 4 5 5 5 5 (15) + 8 = 12 + 8 = 20 x+ 8 = EXAMPLE 1 Solve an equation with a variable on one side Write original equation. Subtract 8 from each side. x = 15 Simplify.

  12. Restaurant During one shift, a waiter earns wages of $30 and gets an additional 15% in tips on customers’ food bills. The waiter earns $105. What is the total of the customers’ food bills? Write a verbal model. Then write an equation. Write 15% as a decimal. EXAMPLE 2 Write and use a linear equation SOLUTION

  13. ANSWER The total of the customers’ food bills is $500. EXAMPLE 2 Write and use a linear equation 105 = 30 + 0.15x Write equation. 75 = 0.15x Subtract 30 from each side. 500 = x Divide each side by 0.15.

  14. ANSWER The correct answer is D EXAMPLE 3 Standardized Test Practice SOLUTION 7p + 13 = 9p – 5 Write original equation. 13 = 2p – 5 Subtract 7pfrom each side. 18 = 2p Add 5 to each side. 9 = p Divide each side by 2.

  15. 7(9) + 13 9(9) – 5 63 + 13 81 – 5 ? ? = = EXAMPLE 3 Standardized Test Practice CHECK 7p+ 13 = 9p– 5 Write original equation. Substitute 9 for p. Multiply. 76 = 76 Solution checks.

  16. x = 2 2 ANSWER The solution 5 5 EXAMPLE 4 Solve an equation using the distributive property Solve 3(5x – 8) = – 2(– x + 7) – 12x. 3(5x – 8) = – 2(– x + 7) – 12x Write original equation. 15x – 24 = 2x – 14 – 12x Distributive property 15x – 24 = – 10x – 14 Combine like terms. 25x – 24 = –14 Add 10xto each side. 25x = 10 Add 24 to each side. Divide each side by 25 and simplify.

  17. 3 (5 – 8) – 2(– +7) – 12 24 3( – 6) –14 – 5 4 5 Substitute for x. 2 2 2 5 5 5 2 ? ? = = 5 EXAMPLE 4 Solve an equation using the distributive property CHECK Simplify. – 18 = – 18 Solution checks.

  18. Solve the formula for r. STEP 1 = r STEP 2 Substitute the given value into the rewritten formula. C 7 r = = 2π 44 C 2π 2π ANSWER The radius of the circle is about 7 inches. EXAMPLE 1 Rewrite a formula with two variables Solve the formula C =2πrfor r. Then find the radius of a circle with a circumference of 44 inches. SOLUTION C = 2 π r Write circumference formula. Divide each side by 2π. Substitute 44 for Cand simplify.

  19. Solve the formula P = 2l + 2wfor w. Then find the width of a rectangle with a length of 12 meters and a perimeter of 41 meters. Solve the formula for w. STEP 1 P – 2l = w 2 EXAMPLE 2 Rewrite a formula with three variables SOLUTION P = 2l + 2w Write perimeter formula. P – 2l = 2w Subtract 2l from each side. Divide each side by 2.

  20. w = ANSWER The width of the rectangle is 8.5 meters. 41 – 2(12) 2 STEP 2 Substitute the given values into the rewritten formula. EXAMPLE 2 Rewrite a formula with three variables Substitute 41 for Pand 12 for l. w = 8.5 Simplify.

  21. Solve the equation for y. STEP 1 x y = + – 9 7 4 4 EXAMPLE 3 Rewrite a linear equation Solve 9x – 4y = 7 for y. Then find the value ofywhen x= –5. SOLUTION 9x – 4y = 7 Write original equation. – 4y = 7 – 9x Subtract 9xfrom each side. Divide each side by –4.

  22. (–5) 9(–5) – 4(– 13) 7 45 – y = – 4 y = + – 7 7 9 4 4 4 ? = EXAMPLE 3 Rewrite a linear equation Substitute the given value into the rewritten equation. STEP 2 Substitute –5 forx. Multiply. y = – 13 Simplify. CHECK 9x– 4y= 7 Write original equation. Substitute –5 for xand –13 for y. 7 = 7 Solution checks.

  23. Solve the equation for y. STEP 1 6 y = 2 + x EXAMPLE 4 Rewrite a nonlinear equation Solve 2y + xy = 6 for y. Then find the value of ywhen x= –3. SOLUTION 2y + x y = 6 Write original equation. (2+ x) y = 6 Distributive property Divide each side by ( 2 + x).

  24. Substitute the given value into the rewritten equation. STEP 2 6 y = 2 + (– 3) EXAMPLE 4 Rewrite a nonlinear equation Substitute –3 for x. y = –6 Simplify.

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