Math Problem Statement
Of two numbers, 4 times the smaller one is less than 3 times the larger one by 6. if the sum of the numbers is larger than 6 times their difference by 5, find the number.
Solution
Let's denote the two numbers as (the smaller number) and (the larger number). We are given the following information:
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Condition 1: "4 times the smaller one is less than 3 times the larger one by 6." This translates to:
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Condition 2: "The sum of the numbers is larger than 6 times their difference by 5." This translates to:
Step 1: Simplify Condition 2
Rewrite Condition 2 for clarity: Expanding and rearranging terms, we get: Bringing terms involving and to one side, we get:
So now, we have two equations:
Step 2: Solve the System of Equations
Substitute for from the first equation
From the first equation: Solving for , we get:
Substitute in the second equation
Now, substitute into the second equation: Expanding and simplifying: Multiply every term by 3 to clear the fraction:
Substitute back to find
Now substitute into :
Solution
The two numbers are:
Verification
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Condition 1:
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Condition 2:
Both conditions are satisfied. Therefore, the solution is correct:
Would you like a further breakdown of any part?
Here are some related questions to extend this topic:
- How do we solve systems of linear equations with more than two unknowns?
- Can you provide an example of solving a system of equations by substitution versus elimination?
- What are some real-world applications of solving linear equations in two variables?
- How can inequalities be used alongside equations in system solutions?
- What is the difference between dependent and independent systems of equations?
Tip: When solving systems of equations, check if substitution or elimination is more straightforward based on how the equations are structured.
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Formulas
4x = 3y - 6
x + y = 6(y - x) + 5
Theorems
Linear System Solution Techniques
Suitable Grade Level
Grades 9-10
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