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 .
- Let the second number be .
From the problem, we derive the following relationships:
- (The first number is 2 less than the second.)
- (Twice the second number is 2 less than 3 times the first.)
Step 2: Substitute into
Replace in the second equation:
Step 3: Simplify the Equation
Expand and simplify:
Rearrange terms:
Step 4: Solve for
Using :
Final Answer:
The two numbers are:
Verification:
- : (Correct)
- : ; (Correct)
Let me know if you have any questions!
Related Questions:
- How would the solution change if the first number were 3 less than the second?
- Can you solve this problem graphically?
- What happens if we reverse the roles of the numbers in the second equation?
- How do you represent the problem in matrix form?
- 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.
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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
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