Math 015 Section 6.4 Translating Equations

Translating Sentences into Equations An equation states that two mathematical expressions are equal To translate a sentence into an equation requires recognition of the words or phrases that mean “equals” equals is is equal to amounts to represents = Once the sentence is translated into an equation, the equation can be solved by rewriting into a form using variables and constants. Example: Sentence: five less than a number is thirteen Equation: five less than a number = thirteen n – 5 = 13 n = 18

Assigning variable expressions Problem: Eight less than three times a number equals five times the number. Find the number. 8 less than 3 times a number = 5 times the number Let: x = the number Assigning variables to the things involved in the problem should be shown as part of your work. 3x – 8 = 5x -8 = 2x -4 = x Answer: The number is -4

Problem: Translate “four more than five times a number is six less than three times the number” into an equation and then solve the equation. 4 more than 5 times a number = 6 less than 3 times the number Let: x = the number 5x + 4 = 3x – 6 5x = 3x – 10 2x = -10 x = -5 Answer: The number is -5

Answer: There are 2,000,000 first-year students at U.S. Colleges Problem: Six hundred thousand of the first-year students at U.S. colleges are taking at least one remedial class in reading, writing, or mathematics. This is three-tenths of the total number of first-year students. How many first year students are in U.S. colleges. 600,000 is 3/10 of the first year students Let: x = the number of 1st year students. 10 3 10 600,000 = x 6,000,000 = 3x 2,000,000 = x Answer: There are 2,000,000 first-year students at U.S. Colleges

Let: x = # of rolls of wallpaper uses Problem: A wallpaper hanger charges $50 plus $28 per roll of wallpaper used in a room. If the total charge for hanging wallpaper in a room is $218, how many rolls of wallpaper were used in that room? Price = $50 + $28 times the # of rolls Let: x = # of rolls of wallpaper uses 218 = 50 + 28x 168 = 28x 6 = x Answer: 6 rolls of wallpaper were used for theroom.

Answer: Shorter piece = 6 ft Longer piece = 14 ft 5x = -2x + 42 Problem: A board that is 20 ft long is cut into two pieces. Five times the length of the shorter piece is 2 ft more than twice the length of the longer piece. Find the length of each piece. x (20 – x) 20 ft Let: x = short piece 20 – x = long piece 5 times the short piece 2 times the long piece = + two 5x = 2(20 – x) + 2 5x = 40 – 2x + 2 Answer: Shorter piece = 6 ft Longer piece = 14 ft 5x = -2x + 42 7x = 42 x = 6

5 less than 3 times the larger # Problem: The sum of two numbers is 9. Eight times the smaller numbers is five less than three times the larger number. Find the two numbers. 8 times the small # = 5 less than 3 times the larger # Let: x = the smaller number 9 – x = the larger number 8x = 3(9 – x) – 5 8x = 27 – 3x – 5 8x = -3x + 22 11x = 22 x = 2 Answer: The smaller number is 2 The larger number is 7