The difference between two numbers is 9. The first number plus twice the other number is 27. Find the two numbers.
Understand the Problem
The question is asking to find two numbers based on two conditions: the difference between the two numbers is 9, and the sum of the first number and twice the second number is 27. We will set up equations based on these conditions and solve for the two numbers.
Answer
The two numbers are $15$ and $6$.
Answer for screen readers
The two numbers are $15$ and $6$.
Steps to Solve
- Define the Variables
Let the first number be $x$ and the second number be $y$.
- Set Up the Equations
Based on the problem:
From the first condition (difference): $$ x - y = 9 $$ From the second condition (sum): $$ x + 2y = 27 $$
- Solve for One Variable
From the first equation, we can express $x$ in terms of $y$: $$ x = y + 9 $$
- Substitute and Solve
Substitute $x$ in the second equation: $$ (y + 9) + 2y = 27 $$
Combine like terms: $$ 3y + 9 = 27 $$
- Isolate the Variable
Subtract 9 from both sides: $$ 3y = 18 $$
Then divide by 3: $$ y = 6 $$
- Find the First Number
Now substitute $y$ back into the equation for $x$: $$ x = 6 + 9 = 15 $$
The two numbers are $15$ and $6$.
More Information
The first number ($15$) is greater than the second number ($6$) by $9$, and the first number plus twice the second number equals $27$.
Tips
- Forgetting to correctly isolate one of the variables before substituting can lead to wrong results.
- Not carefully combining like terms may also cause errors in solving the equations.
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