Lesson 2-5 Warm-Up

“Equations and Problem Solving” (2-5) How do you define one variable in terms of another? What are consecutive integers, and how do you solve consecutive integer problems? Tip: Some problems will contain two or more unknown quantities. To solve such problems, first decide which unknown quantity the variable will represent. Then, express the other unknown quantity or quantities in terms of that variable. Example: The length of a rectangle is 6 in. longer than its width. What is the length of the rectangle in terms of the width? Define: Let w = the width. Then, w + 6 = the length. Consecutive Integers: integers (positive or negative numbers) that are written in order (differ by one from one integer to the next) Consecutive integer problems are example of problems that must be solved by expressing some unknown quantities (the integers after the first one) in terms of the first unknown quantity (first integer). Example: The sum of three consecutive integers is 147. Find the integers. Define: Let n = the first integer. Then, n + 1= the second integer and n + 1= the second integer. Equation: (n) + (n + 1) + (n + 2) = 147 3n + 3 = 147; 3n = 144; n = 48, n + 1 = 49, n + 2 = 50 The length is described in terms of the width, so define the variable for the width first.

Words: The width is 3 in. less than the length. Equations and Problem Solving LESSON 2-5 Additional Examples The width of a rectangle is 3 in. less than its length. The perimeter of the rectangle is 26 in. What is the width of the rectangle? Words: The width is 3 in. less than the length. Define: Let x = the length. The width is described in terms of the length. So define a variable for the length first. Then x – 3 = the width. Equation: P = 2 + 2w Use the perimeter formula. 26 = 2 x + 2( x – 3 ) Substitute 26 for P, x for , and x – 3 for w.

26 = 2x + 2x – 6 Use the Distributive Property. Equations and Problem Solving LESSON 2-5 Additional Examples (continued) 26 = 2x + 2(x – 3) 26 = 2x + 2x – 6 Use the Distributive Property. 26 = 4x – 6 Combine like terms. 26 + 6 = 4x – 6 + 6 Add 6 to each side. 32 = 4x Simplify. = Divide each side by 4. 32 4 4x 4 8 = x Simplify. The width of the rectangle is 3 in. less than the length, which is 8 in. So the width of the rectangle is 5 in.

The sum of three consecutive integers is 72. Find the integers. Equations and Problem Solving LESSON 2-5 Additional Examples The sum of three consecutive integers is 72. Find the integers. Words: first plus second plus third is 72 integer integer integer Define: Let x = the first integer. Then x + 1 = the second integer, and x + 2 = the third integer. Equation: x + x + 1 + x + 2 = 72

3x + 3 – 3 = 72 – 3 Subtract 3 from each side. Equations and Problem Solving LESSON 2-5 Additional Examples (continued) 1x + 1x + 1 + 1x + 2 = 72 3x + 3 = 72 Combine like terms. 3x + 3 – 3 = 72 – 3 Subtract 3 from each side. 3x = 69 Simplify. 3x 3 69 3 = Divide each side by 3. x = 23 Simplify. If x = 23, then x + 1 = 24, and x + 2 = 25. The three integers are 23, 24, and 25.

“Equations and Problem Solving” (2-5) What is uniform motion? How do you solve problems involving distance, rate (like speed), and time? uniform motion: an object that moves at a uniform, or constant (unchanging) rate of speed Rule (Formula): The relationship between distance, rate, and time is given by: d = r t where d = distance, r = rate, and t = time Uniform motion problems may involve objects going in the same direction, opposite directions, or round trips. Example: In the diagram below, the two vehicles are traveling the same direction at different rates. By using the formula d = rt, we determine that both cars will drive a distance of 200 miles. 40 mi. x 5 h. = 200 mi. = 200 mi. 1 h. 1 1 50 mi. x 4 h. = 200 mi. = 200 mi. 1 h. 1 1

Define: Let t = the time the airplane travels. Equations and Problem Solving LESSON 2-5 Additional Examples An airplane left an airport flying at 180 mi/h. A jet that flies at 330 mi/h left 1 hour later. The jet follows the same route as the airplane at a different altitude. How many hours will it take the jet to catch up with the airplane? Define: Let t = the time the airplane travels. Then t – 1 = the time the jet travels. Aircraft Rate Time Distance Traveled Airplane 180 t 180t Jet 330 t – 1 330(t – 1)

Words: distance traveled equals distance traveled by airplane by jet Equations and Problem Solving LESSON 2-5 Additional Examples (continued) Words: distance traveled equals distance traveled by airplane by jet Write: 180 t = 330( t – 1 ) 180t = 330(t – 1) 180t = 330t – 330 Use the Distributive Property. 180t – 330t = 330t – 330 – 330t Subtract 330t from each side. –150t = –330 Combine like terms. –150t –150 –330 = Divide each side by –150. 1 5 t = 2 Simplify. 1 5 t – 1 = 1 The jet will catch up with the airplane in 1 h. 1 5

Define: Let x = time of trip uphill. Equations and Problem Solving LESSON 2-5 Additional Examples Suppose you hike up a hill at 4 km/h. You hike back down at 6 km/h. Your hiking trip took 3 hours. How long was your trip up the hill? Define: Let x = time of trip uphill. Then 3 – x = time of trip downhill. Words: distance uphill equals distance downhill Part of hike Rate Time Distance hiked Uphill 4 x 4x Downhill 6 3 – x 6(3 – x) Equation: 4 x = 6( 3 – x )

4x = 18 – 6x Use the Distributive Property. Equations and Problem Solving LESSON 2-5 Additional Examples (continued) 4x = 6(3 – x) 4x = 18 – 6x Use the Distributive Property. 4x + 6x = 18 – 6x + 6x Add 6x to each side. 10x = 18 Combine like terms. = Divide each side by 10. 10x 10 18 x = 1.8 Simplify. Your trip uphill was 1.8 h long.

:Define Let x = the speed of the jet flying east. Equations and Problem Solving LESSON 2-5 Additional Examples Two jets leave San Francisco at the same time and fly in opposite directions. One is flying west 50 mi/h faster than the other. After 2 hours, they are 2500 miles apart. Find the speed of each jet. :Define Let x = the speed of the jet flying east. Then x + 50 = the speed of the jet flying west. Words: eastbound jet’s plus westbound jet’s equals the total distance distance distance Jet Rate Time Distance Traveled Eastbound x 2 2x Westbound x + 50 2 2(x + 50) Write: 2 x + 2( x + 50 ) = 2500

2x + 2x + 100 = 2500 Use the Distributive Property. Equations and Problem Solving LESSON 2-5 Additional Examples (continued) 2x + 2(x + 50) = 2500 2x + 2x + 100 = 2500 Use the Distributive Property. 4x + 100 = 2500 Combine like terms. 4x + 100 – 100 = 2500 – 100 Subtract 100 from each side. 4x = 2400 Simplify. = Divide each side by 4. 4x 4 2400 x = 600 x + 50 = 650 The jet flying east is flying at 600 mi/h. The jet flying west is flying at 650 mi/h.

The sum of three consecutive integers is 117. Find the integers. Equations and Problem Solving LESSON 2-5 Lesson Quiz The sum of three consecutive integers is 117. Find the integers. You and your brother started biking at noon from places that are 52 mi apart. You rode toward each other and met at 2:00 p.m. Your brother’s average speed was 4 mi/h faster than your average speed. Find both speeds. 3. Joan ran from her home to the lake at 8 mi/h. She ran back home at 6 mi/h. Her total running time was 32 minutes. How much time did it take Joan to run from her home to the lake? 38, 39, 40 your speed: 11 mi/h; brother’s speed: 15 mi/h about 13.7 minutes