STEP 2 Substitute the given value into the rewritten formula. The radius of the circle is about 7 inches.
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1 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 = 2 π r C 2π = r Write circumference formula. Divide each side by 2π. STEP 2 Substitute the given value into the rewritten formula. r = C 2π = 44 2π 7 Substitute 44 for C and simplify. ANSWER The radius of the circle is about 7 inches.
2 GUIDED PRACTICE for Example 1 1. Find the radius of a circle with a circumference of 25 feet. SOLUTION STEP 1 Solve the formula for r. C = 2 π r C 2π = r Write circumference formula. Divide each side by 2π. STEP 2 Substitute the given value into the rewritten formula. r = C 2π = 25 2π 3.9 Substitute 25 for C and simplify. ANSWER The radius of the circle is about 4 feet.
3 GUIDED PRACTICE for Example 1 2. The formula for the distance d between opposite vertices of a regular hexagon is d = 2a where a is 3 the distance between opposite sides. Solve the formula for a. Then find a when d = 10 centimeters. SOLUTION a = d 3 2 When d = 10cm, a = 5 3 cm or 8.7cm
4 EXAMPLE 2 Rewrite a formula with three variables 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. SOLUTION STEP 1 Solve the formula for w. P = 2l + 2w P 2l = 2w Write perimeter formula. Subtract 2l from each side. P 2l 2 = w Divide each side by 2.
5 EXAMPLE 2 Rewrite a formula with three variables STEP 2 Substitute the given values into the rewritten formula. w = w = (12) 2 Substitute 41 for P and 12 for l. Simplify. ANSWER The width of the rectangle is 8.5 meters.
6 GUIDED PRACTICE for Example 2 3. Solve the formula P = 2l + 2w for l. Then find the length of a rectangle with a width of 7 inches and a perimeter of 30 inches. STEP 1 Solve the formula for p = 2l + 2w for l P = 2l + 2w P 2w = 2l P 2w 2 = l Write perimeter formula. Subtract 2w from each side. Divide each side by 2
7 GUIDED PRACTICE for Example 2 STEP 2 Substitute the given values in. Formula for l = P 2w 2 = = 30 2 (7) Multiply. Subtract. = 8 Divide. ANSWER Length of rectangle is 8 in.
8 GUIDED PRACTICE for Example 2 4. Solve the formula A = lw for w. Then find the width of a rectangle with a length of 16 meters and an area of 40 square meters. STEP 1 Solve the formula for w A = lw Write perimeter formula. A l = w Divide each side by l
9 GUIDED PRACTICE for Example 2 STEP 2 Substitute the given values into rewrite formula. w = A Write perimeter formula. l w = Subtract 40 from A and 16 for l. w = 2.5 Divide Write of rectangle is 2.5 m
10 GUIDED PRACTICE for Example 2 Solve the formula for the variable in red. Then use the given information to find the value of the variable. 5. A = 1 2 bh Find h if b = 12 m and A = 84 m 2. A = 1 2 bh Write perimeter formula. 2A = bh Multiply each side by 2. 2A = h b Divide each side by b
11 GUIDED PRACTICE for Example 2 Find the value of h if b = 12m and A = 84m 2. 2A = h b Find h if b = 12 m and A = 84 m 2. 2A = h b 2(84) = h 12 h = 14 Write formula. Substitute 84 for A and 12 for b. Simplify h = 14m
12 GUIDED PRACTICE for Example 2 Solve the formula for the variable in red. Then use the given information to find the value of the variable. 6. A = 1 2 bh Find b if h = 3 cm and A = 9 cm 2. A = 1 2 bh Write perimeter formula. 2A = bh Multiply each side by 2. 2A = b h Divide each side by h
13 GUIDED PRACTICE for Example 2 Solve the formula for the variable in red. Then use the given information to find the value of the variable. 6. A = 1 2 bh Find b if h = 3 cm and A = 9 cm 2. 2A = b h Write formula. 2(9) 3 = b Substitute 9 for A and 3 for b. b = 6 Simplify b = 6cm
14 GUIDED PRACTICE for Example 2 Solve the formula for the variable in red. Then use the given information to find the value of the variable. 7. A = 1 (b b 2 )h Find h if b 1 = 6 in., b 2 = 8 in., and A = 70 in. 2 A = 1 2 (b 1 + b 2 )h 2A = (b 1 + b 2 )h 2A h = (b 1 + b 2 ) Write perimeter formula. Multiply each side by 2. Divide by (b 1 + b 2 )
15 GUIDED PRACTICE for Example 2 Solve the formula for the variable in red. Then use the given information to find the value of the variable. 7. A = 1 (b b 2 )h Find h if b 1 = 6 in., b 2 = 8 in., and A = 70 in. 2 2A h = (b 1 + b 2 ) h = 2(70) (6 + 8) Write formula. Substitute 70 for A and 6 for b 1 and 8 for b 2. h = 10 Simplify h = 10 in.
16 EXAMPLE 3 Rewrite a linear equation Solve 9x 4y = 7 for y. Then find the value of y when x = 5. SOLUTION STEP 1 Solve the equation for y. 9x 4y = 7 4y = 7 9x y = x Write original equation. Subtract 9x from each side. Divide each side by 4.
17 EXAMPLE 3 Rewrite a linear equation STEP 2 Substitute the given value into the rewritten equation. y = ( 5) Substitute 5 for x. 4 4 y = 7 45 Multiply. 4 4 y = 13 CHECK 9x 4y = 7? 9( 5) 4( 13) = 7 7 = 7 Simplify. Write original equation. Substitute 5 for x and 13 for y. Solution checks.
18 EXAMPLE 4 Rewrite a nonlinear equation Solve 2y + xy = 6 for y. Then find the value of y when x = 3. SOLUTION STEP 1 Solve the equation for y. 2y + x y = 6 (2+ x) y = 6 y = x Write original equation. Distributive property Divide each side by ( 2 + x).
19 EXAMPLE 4 Rewrite a nonlinear equation STEP 2 y = ( 3) Substitute the given value into the rewritten equation. Substitute 3 for x. y = 6 Simplify.
20 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = y 6x = 7 STEP 1 y 6x = 7 STEP 2 Solve the equation for y. y = 7 + 6x Substitute the given value into the rewritten equation. y = (2) Substitute 2 for n. y = y = 19 Multiply. Add.
21 GUIDED PRACTICE for Examples 3 and 4 CHECK y 6x = 7? 19 6 (2) = 7 7 = 7 Write original equation. Substitute 2 for x and 19 for y. Solution checks.
22 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = y x = 13 STEP 1 Solve the equation for y. 5y x = 13 5y = 13 + x y = x 5 Write original equation Add x to each side Divide each side by 5
23 GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute the given value into the rewritten equation. 13 y = 5 y = Substitute 2 for n. Simplify. CHECK 5y x = 13? 5(5) 2 = 7 3 = 3 Write original equation. Substitute 2 for x and 19 for y. Solution checks.
24 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = x + 2y = 12 STEP 1 Solve the equation for y. 3x + 2y = 12 5y = 12 3x y = x 2 = 3x Write original equation Subtract 3x from each side Divide each side by 2
25 GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute the given value into the rewritten equation. y = 3 y = 2 2 y = 3 (2) Substitute 2 for n. Multiply Simplify CHECK 3x yx = 12? 3(2) + 2(3) = = 12 Write original equation. Substitute 3 for y and 2 for x. Solution checks.
26 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = x + 5y = 1 STEP 1 Solve the equation for y. 2x + 5y = 1 Write original equation 5y = 1 2x y = = x 5 2x 5 Subtract 2x from each side Divide each side by 5
27 GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute the given value into the rewritten equation. y = 2 1 y = 5 1 y = 1 2(2) Substitute 2 for n. Multiply Simplify CHECK 2x + 5y = 1? 2(2) + 5( 1) = 1 1 = 1 Write original equation. Substitute 2 for x and 1 for y. Solution checks.
28 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = = 2xy x STEP 1 Solve the equation for y. 3 = 2xy x 3 + x = 2xy Write original equation Add x to each side 3 +x 2x = y Divide 2x to each side
29 GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute the given value into the rewritten equation (2) 5 y = 4 = y = Substitute 2 for n. Simplify CHECK 3= 2xy x? 3 = 2(2) ( 5 (2) 4 ) 3 = 3 Write original equation. Substitute 2 for x and 5 1 for y. 4 Solution checks.
30 GUIDED PRACTICE for Examples 3 and 4 Solve the equation for y. Then find the value of y when x = y xy = 28 STEP 1 Solve the equation for y. 4y xy = 28 (4 x)y = x = y Write original equation Distributive property Divide each side by (4 x)
31 GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute the given value into the rewritten equation. y = Substitute 2 for x. y = 14 Simplify CHECK 4y xy = = 28 Write original equation.? 4(14) (2) (14) = 28 Substitute 14 for y and 2 for y. Solution checks.
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