Fix the mistake: Applications of Linear Equations in One Variable

Using Equations ① The Process for Solving Word Problems

Equations are powerful tools for solving word problems. If you read the problem carefully and determine what to represent with the letter x and what quantities are equal, you can solve most word problems using equations.

Five Steps for Solving Word Problems with Equations

  1. Let x represent the quantity you want to find (or a related quantity)
  2. Express the remaining quantities using x
  3. Find an equality relationship and form an equation
  4. Solve the equation to find the value of x
  5. Check whether the solution satisfies the conditions of the problem (Negative numbers of objects or excessively large answers are not acceptable.)
Example ①
When a certain number is multiplied by 4 and then 5 is subtracted, the result is 13 greater than the original number. Find the original number.
Solution:
① Let the original number be x.
② The number obtained by multiplying by 4 and subtracting 5 = 4x − 5; the number 13 greater than the original number = x + 13.
③ Equality relationship: 4x − 5 ＝ x + 13
④ Solve: 4x − x = 13 + 5 → 3x = 18 → x = 6
⑤ Check: 4×6 − 5 = 19, and 6 + 13 = 19. The answer is correct.
　The original number is 6.

Tips for Choosing What to Represent with a Letter

The first thing that often causes difficulty in a word problem is deciding what to represent with x.
　・Basically, let x represent what the problem asks you to find.
　・If it is difficult to form an equation, you may let x represent a related quantity instead.
　・In problems involving a total, such as “in all” or “combined”, let one quantity be x and the other be “(total)−x”.

Example ②
You buy a total of 10 oranges at 80 yen each and apples at 120 yen each, and the total is exactly 1000 yen. How many oranges did you buy?
Solution:
① Let the number of oranges be x.
② Number of apples = 10 − x (apples)
③ Total cost: 80x + 120(10 − x) = 1000
④ Solve: 80x + 1200 − 120x = 1000 → −40x = −200 → x = 5
⑤ Check: 5 oranges and 5 apples. Cost 80×5 + 120×5 = 400 + 600 = 1000. The answer is correct.
　You bought 5 oranges.
Key Points
  • Word problems follow five steps: “Let x represent a quantity → express it with an equation → form an equality → solve → check.”
  • When forming an equation, find “which two quantities are equal” in the problem.
  • In problems involving a total or a difference, the other quantity can be expressed as “the other quantity = total − x”.
  • Always check whether the answer fits the problem (for example, quantities must be nonnegative integers).

Practice Problems

  1. The number obtained by adding 4 to three times a certain number is equal to five times the original number. Find the number.
  2. If 4 candies are given to each child, 3 candies remain; if 5 candies are given to each child, there are 2 candies too few. Find the number of children.
  3. You buy a total of 12 notebooks at 150 yen each and pens at 80 yen each, for a total of 1240 yen. How many notebooks did you buy?

Answers and Explanations

  1. Answer: 2
    Explanation: Let the number be x. 3x + 4 = 5x → 2x = 4 → x = 2.
  2. Answer: 5 children
    Explanation: Let the number of children be x. The number of candies can be expressed as either 4x+3 or 5x−2. 4x + 3 = 5x − 2 → x = 5.
  3. Answer: 4 notebooks
    Explanation: Let the number of notebooks be x and the number of pens be (12−x). 150x + 80(12−x) = 1240 → 70x + 960 = 1240 → 70x = 280 → x = 4.

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