Math Problem Statement

One number is 2 less than a second number. Twice the second number is 2 less than 3 times the first. Find the two numbers.

Solution

To solve the problem, let us define the numbers mathematically:

Step 1: Define Variables

  • Let the first number be xx.
  • Let the second number be yy.

From the problem, we derive the following relationships:

  1. x=y2x = y - 2 (The first number is 2 less than the second.)
  2. 2y=3x22y = 3x - 2 (Twice the second number is 2 less than 3 times the first.)

Step 2: Substitute x=y2x = y - 2 into 2y=3x22y = 3x - 2

Replace xx in the second equation: 2y=3(y2)22y = 3(y - 2) - 2

Step 3: Simplify the Equation

Expand and simplify: 2y=3y622y = 3y - 6 - 2 2y=3y82y = 3y - 8

Rearrange terms: 2y3y=82y - 3y = -8 y=8-y = -8 y=8y = 8

Step 4: Solve for xx

Using x=y2x = y - 2: x=82x = 8 - 2 x=6x = 6

Final Answer:

The two numbers are: x=6 and y=8\boxed{x = 6 \text{ and } y = 8}

Verification:

  1. x=y2x = y - 2: 6=826 = 8 - 2 (Correct)
  2. 2y=3x22y = 3x - 2: 2(8)=3(6)22(8) = 3(6) - 2; 16=18216 = 18 - 2 (Correct)

Let me know if you have any questions!


Related Questions:

  1. How would the solution change if the first number were 3 less than the second?
  2. Can you solve this problem graphically?
  3. What happens if we reverse the roles of the numbers in the second equation?
  4. How do you represent the problem in matrix form?
  5. What are some real-world applications of such problems?

Tip:

When solving word problems, always start by defining clear variables and translating the conditions into equations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

Substitution method: Substitute one variable's value into another equation
Rearranging linear equations to isolate variables

Theorems

Properties of equality: Operations can be performed equally on both sides of an equation

Suitable Grade Level

Grades 6-8