1. Math 015 Section 6.4 Translating Equations

2. 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

3. 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

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

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

6. 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.

7. 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

8. 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