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.

Let: x = the number Assigning variables to the things involved in the problem should be shown as part of your work.

8 less than 3 times a number = 5 times the number

3x – 8

-8 = 2x

-4 = x Answer:

The number is -4

= 5x

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.

Answer: The number is -5

Let: x = the number

4 more than 5 times a number = 6 less than 3 times the number

5x + 4

5x = 3x – 10

2x = -10

= 3x – 6

x = -5

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

600,000

6,000,000 = 3x

2,000,000 = x

Answer: There are 2,000,000 first-year students at U.S. Colleges

Let: x = the number of 1st year students.

= x 3 10

10

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?

Let: x = # of rolls of wallpaper uses

Price $28 times the # of rolls = $50

168 = 28x

Answer: 6 rolls of wallpaper were used for theroom.

+

218 = 50 + 28x

6 = x

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.

20 ft

x (20 – x)

5x

5x = 40 – 2x + 2

5x = -2x + 42

7x = 42

x = 6

Answer: Shorter piece = 6 ft Longer piece = 14 ft

5 times the short piece

= 2 times the long piece

+ two

Let: x = short piece

20 – x = long piece

= 2(20 – x) + 2

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 #

Answer: The smaller number is 2

The larger number is 7

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