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Sentences as Single Variable Equations

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Algebraic Equations to Represent Words

The sum of two consecutive even integers is 34. What are the integers?

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Khan Academy Problem Solving Word Problems 2

Guidance

To translate a problem from words into an equation, look for key words to indicate the operation used in the problem.

Once the equation is known, to solve the problem you use the same rules as when solving equations with one variable. Isolate the variable and then solve for it making sure that whatever you do to one side of the equals sign you do to the other side. Drawing a diagram is also helpful in solving some word problems.

Example A

Two consecutive integers have a sum of 173. What are those numbers?

Solution: Let x = integer 1

Then x + 1 = integer 2 (Because they are consecutive, they must be separated by only one number. For example: 1, 2, 3, 4,... all are consecutive.)

Translate the sentence into an equation and solve:

x + (x + 1) &= 173\\x + x + 1 &= 173 && (\text{Remove the parentheses})\\2x + 1 &= 173 && (\text{Combine like terms})\\2x + 1 {\color{red}-1} &= 173 {\color{red}-1} && (\text{Subtract} \ 1 \ \text{from both sides to isolate the variable})\\2x &= 172 && (\text{Simplify})\\\frac{2x}{{\color{red}2}} &= \frac{172}{{\color{red}2}} && (\text{Divide both sides by} \ 2 \ \text{to solve for the variable})\\x &= 86 && (\text{Simplify})

Therefore the first integer is 86 and the second integer is (86 + 1) = 87 . Check: 86 + 87=173 .

Example B

When a number is subtracted from 35, the result is 11. What is the number?

Solution: Let x = the number

Translate the sentence into an equation and solve:

35 - x &= 11\\35 {\color{red}- 35} - x &= 11 {\color{red}- 35} && (\text{Subtract} \ 35 \ \text{from both sides to isolate the variable})\\-x &= -24 && (\text{Simplify})\\\frac{-x}{{\color{red}-1}} &= \frac{-24}{{\color{red}-1}} && (\text{Divide both sides by} \ -1 \ \text{to solve for the variable})\\x &= 24 && (\text{Simplify})

Therefore the number is 24.

Example C

When one third of a number is subtracted from one half of a number, the result is 14. What is the number?

Solution: Let x = number

Translate the sentence into an equation and solve:

\frac{1}{2}x-\frac{1}{3}x=14

You need to get a common denominator in this problem in order to solve it. For this problem, the denominators are 2, 3, and 1. The LCD is 6. Therefore multiply the first fraction by \frac{3}{3} , the second fraction by \frac{2}{2} , and the third number by \frac{6}{6} .

\left({\color{red}\frac{3}{3}}\right) \frac{1}{2}x-\left({\color{red}\frac{2}{2}}\right) \frac{1}{3}x &= \left({\color{red}\frac{6}{6}}\right)14\\\frac{3}{6}x-\frac{2}{6}x &= \frac{84}{6} && (\text{Simplify})

Now that the denominator is the same, the equation can be simplified to be:

3x-2x &= 84\\x &= 84 && (\text{Combine like terms})

Therefore the number is 84.

Concept Problem Revisited

The sum of two consecutive even integers is 34. What are the integers?

Let x = integer 1

Then x + 2 = integer 2 (Because they are even, they must be 2 numbers apart. For example: 2, 4, 6, 8,... are all consecutive even numbers.)

Translate the sentence into an equation and solve:

x + (x + 2) &= 34\\x + x + 2 &= 34 && (\text{Remove the parentheses})\\2x + 2 &= 34 && (\text{Combine like terms})\\2x + 2 {\color{red}-2} &= 34 {\color{red}-2} && (\text{Subtract} \ 2 \ \text{from both sides to isolate the variable})\\2x &= 32 && (\text{Simplify})\\\frac{2x}{{\color{red}2}} &= \frac{32}{{\color{red}2}} && (\text{Divide both sides by} \ 2 \ \text{to solve for the variable})\\x &= 16 && (\text{Simplify})

Therefore the first integer is 16 and the second integer is (16 + 2) = 18 . Note that 16 + 18 is indeed 34.

Vocabulary

Algebraic Equation
An algebraic equation contains numbers, variables, operations, and an equals sign.
Consecutive
The term consecutive means in a row. Therefore an example of consecutive numbers is 1, 2, and 3. An example of consecutive even numbers would be 2, 4, and 6. An example of consecutive odd numbers would be 1, 3, and 5.

Guided Practice

1. What is a number that when doubled would equal sixty?

2. The sum of two consecutive odd numbers is 176. What are these numbers?

3. The perimeter of a square frame is 48 in. What are the lengths of each side?

Answers:

1. The number is 30.

2x &= 60\\\frac{2x}{{\color{red}2}} &= \frac{60}{{\color{red}2}} && (\text{Divide by} \ 2 \ \text{to solve for the variable})\\x &= 30 && (\text{Simplify})

2. The first number is 87 and the second number is (87 + 2) = 89 .

x + (x + 2) &= 176\\x + x + 2 &= 176 && (\text{Remove parentheses})\\2x + 2 &= 176 && (\text{Combine like terms})\\2x+2 {\color{red}-2} &= 176 {\color{red}-2} && (\text{Subtract} \ 2 \ \text{from both sides of the equals sign to isolate the variable})\\2x &= 174 && (\text{Simplify})\\\frac{2x}{{\color{red}2}} &=\frac{174}{{\color{red}2}} && (\text{Divide by} \ 2 \ \text{to solve for the variable})\\x &= 87

3. The side length is 12 inches.

s + s + s + s &= 48 && (\text{Write initial equation with four sides adding to the perimeter})\\4s &= 48 && (\text{Simplify})\\\frac{4s}{{\color{red}4}} &= \frac{48}{{\color{red}4}} && (\text{Divide by} \ 4 \ \text{to solve for the variable})\\s &= 12

Practice

  1. The sum of two consecutive numbers is 159. What are these numbers?
  2. The sum of three consecutive numbers is 33. What are these numbers?
  3. A new computer is on sale for 30% off. If the sale price is $500, what was the original price?
  4. Jack and his three friends are sharing an apartment for the next year while at university (8 months). The rent costs $1200 per month. How much does Jack have to pay if they split the cost evenly?
  5. You are designing a triangular garden with an area of 168 square feet and a base length of 16 feet. What would be the height of the triangular garden shape?
  6. If four times a number is added to six, the result is 50. What is that number?
  7. This week, Emma earned ten more than half the number of dollars she earned last week babysitting. If this week, she earned 100 dollars, how much did she earn last week?
  8. Three is twenty-one divided by the sum of a number plus five. What is the number?
  9. Five less than three times a number is forty-six. What is the number?
  10. Hannah had $237 in her bank account at the start of the summer. She worked for four weeks and now she has $1685 in the bank. How much did Hannah make each week in her summer job?
  11. The formula to estimate the length of the Earth's day in the future is found to be twenty–four hours added to the number of million years divided by two hundred and fifty. In five hundred million years, how long will the Earth's day be?
  12. Three times a number less six is one hundred twenty-six. What is the number?
  13. Sixty dollars was two-thirds the total money spent by Jack and Thomas at the store. How much did they spend total?
  14. Ethan mowed lawns for five weekends over the summer. He worked ten hours each weekend and each lawn takes an average of two and one-half hours. How many lawns did Ethan mow?
  15. The area of a rectangular pool is found to be 280 square feet. If the base length of the pool is 20 feet, what is the width of the pool?
  16. A cell phone company charges a base rate of $10 per month plus 5¢ per minute for any long distance calls. Sandra gets her cell phone bill for $21.20. How many long distance minutes did she use?

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