2-1 Creating and Solving Equations. I will create equations to represent situations and solve them to work out problems in context.

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1 2-1 Creating and Solving Equations I will create equations to represent situations and solve them to work out problems in context. Linear Equations To create an equation to represent a word problem, follow this six-step process: 1. Read the problem carefully, and figure out what you need to find. 2. Analyze the problem. Set up a strategy, and assign a variable to the quantity you need to find. 3. Write the strategy sentences as equations or inequalities (=, <, >,, or ). 4. Solve the equations using the correct operations (+,,, ). 5. Verify that you have correctly answered what you were required to find. 6. Check your answer. Your answer must make the equation true and also must make logical sense in the given context. You ve learned that a linear equation is an equation in which the power of the variable does not exceed 1. A linear equation in one variable (x) can be written in the form ax + b = 0, where a and b are constants. Some word problems can be modeled as linear equations. Let s try a sample problem. First we ll create the equation to represent the problem, and then we ll solve the equation to find the value of the unknown quantity. Example 1: Martin bought a used car with an odometer that read 7,500 miles. (An odometer shows the total number of miles a vehicle has been driven). He drives his car 600 miles each month. After a few months, the odometer shows 9,900 miles. How many months has Martin been driving the car? Activity In this activity, you ll model a real-life situation by writing a linear equation. You ll then solve the equation to find an unknown value. A Broadway musical was attended by a total of 2,838 people, including men, women, and children. The number of women who attended the event was 5 more than 3 times the number of children, while the number of men was 3 less than 4 times the number of women. Write an equation to model this situation. Then, use the equation to find the number of men, women, and children who attended the musical.

2 Part A To begin, write expressions to find the numbers of men, women, and children. Using the expressions you obtained in part A, write an equation to model the situation in terms of the variable x. Part C Use the equation you wrote in part B to find the numbers of men, women, and children who attended the play. Example 2: Match the equation with the word problem. a. x + (x + 2) + (x + 4) = 66 b. x + (x + 1) + (x + 2) = 66 c. 2[(x ) + (x + 2)] = 66 The length of a rectangle is 3 inches more than its width. If the length and width are each increased by 2 inches, the perimeter of the rectangle becomes 66 inches. If x is the width of the rectangle, find its width and length. The sum of three consecutive positive even numbers is 66. The smallest number is x. Find the three numbers. The sum of three consecutive positive numbers is 66. The smallest number is x. Find the three numbers.

3 Quadratic Equations A quadratic equation is one in which the highest power of the variable is 2. A quadratic equation is also referred to as an equation of second degree. It is written in the standard form ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0. Some word problems can be modeled as quadratic equations. Area calculations, for example, commonly involve quadratic equations. The six-step process you used earlier to translate word problems into linear equations also works for quadratic equations. Let s solve an area problem that can be modeled as a quadratic equation. The area of a rectangular field is 48 square meters. The length of the field is 7 meters more than 3 times its width. Find the length and width of the field. Activity In this activity, you ll model a problem as a quadratic equation. You ll then solve the equation to find the unknown value. Alistair has a rectangular picture measuring 8 inches by 7 inches. He makes a frame for the picture, shown by the shaded area in the diagram. The area of the frame is 34 square inches, and the width of the frame is x inches. Use this information to complete the activity. Part A Using the given information, write an equation to model the area of the frame. Which type of equation did you write to model the given scenario, linear or quadratic?

4 Part C Use the equation you created in part A to find the width of the picture frame. Try on your own The area of a triangle is 17.5 square meters. The height of the triangle is 3 meters less than twice its base. The base of the triangle is x meters. Complete the equation that represents this description and fill in the values for the base and height of the triangle. The equation modeling the described triangle is x² + x + = 0. The base of the triangle is meters. The height of the triangle is meters. Exponential Equations Sometimes, a quantity with a certain initial value increases or decreases by a multiple. Situations in which this occurs include population growth, appreciation and depreciation of assets, and growth and decay of bacteria. Such situations can be modeled by exponential equations. In an exponential equation, the exponent is a variable. For example, an exponential equation in which P is the initial value of a quantity, r is the rate of growth or decay, t is the number of times the increase or decrease takes place, and A is the final value, is written as A = P(1 ± r) t. The term (1 + r) represents an increase, and (1 r) represents a decrease. Let s model a real-world problem as an exponential equation. We ll create and solve the equation using the same six-step process we used earlier for linear and quadratic equations.

5 Example: A forest currently has 2,000 trees of a particular species. If the number of trees increases at the rate of 10 percent annually, how many years will it take for the forest to have 2,662 trees of that species? The initial number of trees (P) is 2,000, the rate of increase (r) is 10%, and the final value (A) is 2,662. We need to find out how many years it will take for the forest to have 2,662 trees of that species. Let t be the number of years. The number of trees increases every year, so we will use (1 + r) in the exponential equation: Lesson Activity In this activity, you ll model a problem by writing an exponential equation. You ll then solve the equation to find an unknown value. The population of goldfish in a marine farm is decreasing at the rate of 20 percent per year. The farm started with a goldfish population of 1,000. Part A Create an equation that gives the number of goldfish in the nth year. Using the equation you created in part A, find the number of years it will take to reduce the farm s goldfish population to 640.

6 Rational and Roots Equations We can use rational equations to model a variety of scenarios, including those that involve average rate or average cost. A rational equation involves fractions in which the numerator or the denominator, or both, contains the unknown value. Let s model a word problem with a rational equation and then solve the equation to find the unknown value. Audrey is moving and rents a truck. The truck rental company charges a base rate of $20 per day plus $0.59 per mile. If Audrey s average rental cost per mile is $1.09, how many miles did she drive the truck? We can find the average rental cost by dividing the total cost by the number of miles: Example: Steve is remodeling his home. He plans to increase the area of his square deck by 9 square feet. If the length of each side of the new square deck is 5 feet, find the length of the sides of the original square deck. Recall that all of the sides of a square are equal in length, and its area is equal to the square of the length of the side: The length of the sides of the new deck is 5 feet. Let x be the length of the sides of the original deck. So, the area of the original deck is x 2. Therefore, the area of the new deck is x Let s substitute the known values in the equation: Lesson Activity Claire enrolls in a scuba diving program. The program costs include a one-time registration fee of $75 and $60 for each lesson. In addition, Claire had to purchase scuba equipment costing $275. Use this information to complete the activity. Part A

7 Write an equation that models the average cost per lesson (C) in terms of the number of lessons that Claire takes (x). Use the equation you wrote to find the number of lessons Claire took if the average cost per lesson is $110. Question 2 A ladder is leaning against a wall. The distance between the foot of the ladder and the wall (BC) is 7 meters less than the distance between the top of the ladder and the ground (AB). Use this information to perform the following tasks. Part A Create an equation that models the length of the ladder (l) in terms of x, which is the length in meters of AB. If the length of the ladder is 13 meters, use the equation you wrote to find the distance between the ground and the top of the ladder (AB).

8 Try on your own a. Find the radius (in inches) of a circle whose area is 616 in². b. Together, two pipes can fill a tank in 3 hours. One pipe alone can fill the tank in 5hrs. How long does it take for the other pipe to fill the tank alone? c. The hypotenuse of a right triangle is 25 inches, and its height is 5 inches more than its base. What is the height of the triangle? d. A telemarketer is paid a fixed amount of $15 in addition to $2 per call. If the average cost per call is $3, find the number of calls made by the telemarketer.