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 larger numbers. (do it in simpler way)
Solution
Let's break down the problem and solve it step-by-step in a simpler way.
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Assign Variables:
- Let the two numbers be and , where is the smaller number and is the larger number.
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Set Up Equations Based on the Problem Statements:
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First Statement: "4 times the smaller one is less than 3 times the larger one by 6."
- This translates to:
- This translates to:
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Second Statement: "The sum of the numbers is larger than 6 times their difference by 5."
- First, the difference between and is .
- So, 6 times their difference is .
- We have:
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Solve the Equations Step-by-Step:
Let's start by simplifying the second equation.
Expanding and rearranging: Bringing like terms together:
Now, we have a system of equations:
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Solve the System of Equations:
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Multiply the first equation by 5 and the second equation by 3 to make the coefficients of equal:
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Subtract the first modified equation from the second:
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Substitute into the first equation to find :
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Conclusion:
- The larger number is .
Let me know if you'd like further explanation or if you have any questions.
Related Questions:
- How would the solution change if the first statement used "greater than" instead of "less than"?
- What if both statements were equalities instead of inequalities?
- How do we solve similar problems if given different numbers or coefficients?
- Can we solve this problem graphically by plotting the equations?
- How would the approach change if the sum were exactly equal to 6 times the difference?
Tip: When dealing with word problems, translating each statement into an equation is key.
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Inequalities
Formulas
4x = 3y - 6
x + y = 6(y - x) + 5
Theorems
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Suitable Grade Level
Grades 8-10