EXAMPLE 1 Evaluate powers a 54 5 b

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

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

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.

EXAMPLE 3 Use a verbal model to solve a problem SOLUTION STEP 1 Write: a verbal model. Then write an algebraic expression. Use the fact that profit is the difference between income and expenses. – 3 c – 120 An expression that shows your profit is 3 c – 120.

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

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

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

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.

EXAMPLE 5 Simplify a mathematical model SOLUTION Write a verbal model. Then write an algebraic expression. An expression for the total cost is 0. 8 n + 0. 2(15 – n) = 0. 8 n + 3 – 0. 2 n = (0. 8 n – 0. 2 n) + 3 Distributive property. Group like terms.

EXAMPLE 5 Simplify a mathematical model = 0. 6 n + 3 Combine like terms. ANSWER When n = 5, the total cost is 0. 6(5) + 3 = 3 + 3 = $6.

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

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

EXAMPLE 2 Write and use a linear equation 105 = 30 + 0. 15 x 75 = 0. 15 x 500 = x Write equation. Subtract 30 from each side. Divide each side by 0. 15. ANSWER The total of the customers’ food bills is $500.

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

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

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

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

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

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

EXAMPLE 2 Rewrite a formula with three variables STEP 2 Substitute the given values into the rewritten formula. 41 – 2(12) w= 2 Substitute 41 for P and 12 for l. w = 8. 5 Simplify. ANSWER The width of the rectangle is 8. 5 meters.

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

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

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

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