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

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

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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. Write an expression that shows your profit from selling c candles. Find your profit if you sell 75 candles.

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EXAMPLE 3 Use a verbal model to solve a problem SOLUTION STEP 1Write: 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. 3 c – 120

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EXAMPLE 3 Use a verbal model to solve a problem STEP 2 Evaluate: the expression in Step 1 when c = 75. 3c – 120 = 3(75) – 120 Substitute 75 for c. = 225 – 120 = 105 Subtract. ANSWER Your profit is $ 105. Multiply.

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

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

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EXAMPLE 5 Simplify a mathematical model Digital Photo Printing 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 n of the 15 prints are in large format. Then find the total cost if 5 of the 15 prints are in large format.

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SOLUTION EXAMPLE 5 Simplify a mathematical model 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) Distributive property. = (0.8n – 0.2n) + 3 Group like terms. = 0.8n + 3 – 0.2n

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

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EXAMPLE 1 Solve an equation with a variable on one side Solve 4 5 x + 8 = 20. 4 5 x + 8 = 20 4 5 x = 12 x = (12) 5 4 x = 15 Write original equation. Subtract 8 from each side. Multiply each side by, the reciprocal of. 5 44 5 Simplify. ANSWER The solution is 15. CHECK x = 15 in the original equation. 4 5 4 5 x + 8 = (15) + 8 = 12 + 8 = 20

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EXAMPLE 2 Write and use a linear equation 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? Restaurant SOLUTION Write a verbal model. Then write an equation. Write 15% as a decimal.

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

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

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

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

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

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

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

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

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

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

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

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

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