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Solving a Two-Number Problem Using "Sum" and "More Than Twice"
Apply the seven-step problem-solving strategy to a number problem that combines multiplication and addition in the relationship between two unknowns, producing a three-step equation.
Problem: One number is ten more than twice another. Their sum is one. Find the numbers.
- Read the problem.
- Identify what to find: two numbers.
- Name the unknowns using a single variable. Let = the first number. Because one number is "ten more than twice" the other, the phrase combines two operations: twice signals multiplication by , and ten more than signals adding . The second number is therefore .
- Translate into an equation: Their sum is , so:
- Solve the equation. First, combine like terms :
Subtract from both sides:
Divide both sides by :
Find the second number: .
- Check: Is ten more than twice ?
Is their sum ?
- Answer: The numbers are and .
This example extends earlier two-number problems in two ways. First, the relationship between the unknowns involves two operations — multiplication followed by addition — rather than a simple additive phrase like "five more than." The expression captures both operations in a single algebraic expression. Second, the resulting equation requires three algebraic steps (combining like terms, subtracting, and dividing) rather than two, because combining and into is needed before the inverse operations can isolate the variable.
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