Section 2.2. Applications of Linear Equations

Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities A few important phrases are described below that can give us clues on how to create an equation that can be used to solve the type of problem we will see in this section. • The word “number” often becomes the variable. (I usually replace the word number with an 𝑥, (unless a different variable makes more sense.) (𝒏𝒖𝒎𝒃𝒆𝒓 = 𝒙) • The word 𝒊𝒔 generally is replaced with an equal sign. (𝒊𝒔 =) 𝑦 𝑖𝑠 12 becomes 𝑦 = 12 • Key phrases need to be transformed into expressions. We learned many of the phrases in section 1.1. • Plus, often represents addition 𝐴 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝𝑙𝑢𝑠 5 becomes 𝑥 + 5 • Twice, often represents multiplication 𝑇𝑤𝑖𝑐𝑒 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 2𝑥 • We will transform key phrases into expressions and use the expressions to create equations to solve the problems in this section. Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities Example 1. 40 less than three times a certain number is 110. What is the number? Key words Algebraic translation number 𝑥 40 less than three times a certain number 3𝑥 – 40 is = 110 110 The equation we need solve is: 3𝑥 − 40 = 110 +40 + 40 3𝑥 = 150 3𝑥 150 = 3 3 𝑥 = 50 Answer: The number is 50 Is this correct? Is three times our answer of 50 less 40 equal 110? Check: 3(50) – 40 = 110 150 – 40 = 110 110 = 110 ✓ Example 2. Ten more than twice a number is equal to 5 times a number plus 1. What is the number? Key words “Number” and “a number” Ten more than twice a number Is equal to 5 times a number plus 1 Translation 𝑥 (these both refer to the same number) 2𝑥 + 10 = 5𝑥 + 1 2𝑥 + 10 = 5𝑥 + 1 −2𝑥 − 2𝑥 10 = 3𝑥 + 1 −1 −1 9 = 3𝑥 9 3𝑥 = 3 3 3=𝑥 Answer: The number is 3 (Answer is correct because: 2(3) + 10 = 5(3) + 1 : 16 = 16 ✓ Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities When a problem asks to find consecutive integers, it is customary to define the numbers as follows: 𝐹𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 𝑆𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 1 𝑇ℎ𝑖𝑟𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 2 𝐹𝑜𝑢𝑟𝑡ℎ 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 3 𝐴𝑛𝑑 𝑠𝑜 𝑜𝑛 … Example 3: The sum of three consecutive integers is 123. What are the integers? Key words Translation 𝐹𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 Three consecutive integers 𝑆𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 1 𝑇ℎ𝑖𝑟𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 2 Sum of three consecutive integers (𝑥) + (𝑥 + 1) + (𝑥 + 2) is = 123 123 (𝑥) + (𝑥 + 1) + (𝑥 + 2) = 123 1𝑥 + 1𝑥 + 1 + 1𝑥 + 2 = 123 (I dropped the parenthesis and put a 1 in front of each x to make the algebra easier to follow.) 3𝑥 + 3 = 123 −3 − 3 3x = 120 3𝑥 120 = 3 3 𝑥 = 40 𝐹𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 40, 𝑆𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 1 = 40 + 1 = 41 𝑇ℎ𝑖𝑟𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 2 = 40 + 2 = 42 𝐴𝑛𝑠𝑤𝑒𝑟 𝑡ℎ𝑒 𝑡ℎ𝑟𝑒𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑟𝑒 40, 41 𝑎𝑛𝑑 42 𝐶ℎ𝑒𝑐𝑘 𝑏𝑦 𝑐𝑜𝑛𝑓𝑖𝑟𝑚𝑖𝑛𝑔 𝑡ℎ𝑒 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑎𝑑𝑑 𝑡𝑜 123: 40 + 41 + 42 = 123 ✓ Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities If a problem asks for consecutive even integers the variables will be 𝐹𝑖𝑟𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟: 𝑥 𝑆𝑒𝑐𝑜𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟: 𝑥 + 2 𝑇ℎ𝑖𝑟𝑑 𝑛𝑢𝑚𝑏𝑒𝑟: 𝑥 + 4 𝐹𝑜𝑢𝑟𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟: 𝑥 + 6 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛, 𝑘𝑒𝑒𝑝 𝑎𝑑𝑑𝑖𝑛𝑔 2 𝑎𝑠 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑟𝑒 𝑡𝑤𝑜 𝑎𝑝𝑎𝑟𝑡. Variables will be defined the same way if a problem asks for consecutive odd integers as odd integers are also two apart. Example 4. The sum of three consecutive even integer is 78. What are the numbers? Key words Translation Three consecutive even integers 𝐹𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 𝑆𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 2 𝑇ℎ𝑖𝑟𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 4 Sum of three consecutive even integers (𝑥) + (𝑥 + 2) + (𝑥 + 4) is = 78 78 (𝑥) + (𝑥 + 2) + (𝑥 + 4) = 78 1𝑥 + 1𝑥 + 2 + 1𝑥 + 4 = 78 (I dropped the parenthesis and put a 1 in front of each x to make the algebra easier to follow.) 3𝑥 + 6 = 78 −6 − 6 3x = 72 3𝑥 72 = 3 3 x = 24 𝐹𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 24 𝑆𝑒𝑐𝑜𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 2 = 24 + 2 = 26 𝑇ℎ𝑖𝑟𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑥 + 4 = 24 + 4 = 28 Answer the three numbers are 24, 26 and 28 The numbers in the answer are clearly consecutive even integers. To check that this answer is correct add up the answers to confirm the sum is 78. 24 + 26 + 28 = 78 ✓ Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities The next example involves angles in a triangle. All triangles have 3 angles. The triangle pictures has angles A, B and C. The sum of angles in all triangles is 180 degrees. Example 5. The second angle of a triangle is double the first. The third angle is 40 less than the first. Find the three angles. Key words Both sentences compare an angle to the first angle. In problems in which we are asked to find the angles of a triangle we always let x be the angle that is compared to the other two angles. The second angle of a triangle is double the first The third angle is 40 less than the first Translation 𝐹𝑖𝑟𝑠𝑡 𝑎𝑛𝑔𝑙𝑒 = 𝑥 𝑆𝑒𝑐𝑜𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 = 2𝑥 𝑇ℎ𝑖𝑟𝑑 𝑎𝑛𝑔𝑙𝑒 = 𝑥 − 40 The above table is all I can get directly from the word problem. I need to know that the equation I need is found by adding the three angles and setting them equal to 180. (𝑥) + (2𝑥) + (𝑥 − 40) = 180 1𝑥 + 2𝑥 + 1𝑥 − 40 = 180 4𝑥 − 40 = 180 +40 + 40 4𝑥 = 220 4𝑥 220 = 4 4 𝑥 = 55 First angle: 𝑥 = 55 Second angle: 2𝑥 = 2 ∗ 55 = 110 Third angle: 𝑥 – 40 = 55 – 40 = 15 (𝑛𝑜𝑡𝑖𝑐𝑒 55 + 110 + 15 = 180) Answer: First angle 55 degrees, Second angle 110 degrees, Third angle 15 degrees Check: Second angle is double the first: 110 is double 55 ✓ Third angle is 40 less than the first: 15 = 55 – 40 ✓ Sum of angles is 180: 55 + 110 + 15 = 180 ✓ Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities 𝐿𝑒𝑡 𝑃 = 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑎 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝐿𝑒𝑡 𝑊 = 𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑎 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝐿𝑒𝑡 𝐿 = 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝑃 = 2𝑊 + 2𝐿 Example 6 The perimeter of a rectangle is 46 inches. The length is 1 less than double the width. Find the dimensions. Key Words Translation Perimeter of a rectangle This automatically gives me the variables and formula I need to get started. 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 𝑃 𝑊𝑖𝑑𝑡ℎ = 𝑊 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐿 Perimeter of a rectangle is 46 length is 5 less than double the width Replace L with 2W - 5 𝑃 = 2𝑊 + 2𝐿 = 46 = 2𝑊 + 2𝐿 (𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑃 𝑤𝑖𝑡ℎ 44) 𝐿 = 2𝑊 − 1 46 = 2𝑊 + 2(2𝑊 − 1) 46 = 2𝑊 + 2(2𝑊 − 1) 46 = 2𝑊 + 4𝑊 − 2 46 = 6𝑊 − 22 +2 +2 48 = 6𝑊 8=W 𝑊𝑖𝑑𝑡ℎ: 8 𝑖𝑛𝑐ℎ𝑒𝑠: 𝐿𝑒𝑛𝑔𝑡ℎ: 𝐿 = 2𝑊 – 1 = 2(8) – 1 = 15 Answer: 𝑊𝑖𝑑𝑡ℎ 8 𝑖𝑛𝑐ℎ𝑒𝑠, 𝐿𝑒𝑛𝑔𝑡ℎ 15 𝑖𝑛𝑐ℎ𝑒𝑠 Check: Length equals 1 less than double the width: 𝐿 = 2(8) − 1 = 15✓ Perimeter is 46: 𝑃 = 2𝑊 + 2𝐿 = 2(8) + 2(15) = 16 + 30 = 46 ✓ Section 2.2: Applications of Linear equations Chapter 2: Functions, Linear equations, and inequalities Example 7 A sofa and a love seat together costs S1500. The sofa costs double the love seat. How much does each cost? Key Words Translation Just by reading I can tell I need to find the cost 𝑆 = 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑠𝑜𝑓𝑎 of a sofa and the cost of a love seat. Each cost 𝐿 = 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑙𝑜𝑣𝑒 𝑠𝑒𝑎𝑡 needs a variable. A sofa and a love seat together costs S1500 𝑆 + 𝐿 = 1500 The sofa costs double the love seat 𝑆 = 2𝐿 Replace the S with 2L in the S + L = 1500 2𝐿 + 𝐿 = 1500 equation 2𝐿 + 𝐿 = 1500 3𝐿 = 1500 3𝐿 1500 = 3 3 L = 500 S = 2L S = 2(500) = 1000 Answer: Soft costs $1000, Love seat costs $500. Does this check? Sofa cost double the love seat? Yes $1000 is double $500 ✓ Together they cost $1500, yes $1000 + $500 = $1500 ✓