If 45 is subtracted from twice the greater of two numbers, it results in the other number. If 21 is subtracted from twice the smaller number, it results in the greater number. Find the numbers.

Answer

To solve the problem, we need to set up a system of linear equations based on the information given. Let's denote the two numbers as follows:

- Let \( x \) be the greater number.
- Let \( y \) be the smaller number.

Now, we can translate the statements from the problem into equations.

### Step 1: Formulate the equations

1. According to the first statement, "If 45 is subtracted from twice the greater of two numbers, it results in the other number," we can write the equation:
\[ 2x - 45 = y \quad \text{(Equation 1)} \]

2. According to the second statement, "If 21 is subtracted from twice the smaller number, it results in the greater number," we can write the equation:
\[ 2y - 21 = x \quad \text{(Equation 2)} \]

### Step 2: Solve the equations

Now we have a system of two equations:
1. \( 2x - 45 = y \)
2. \( 2y - 21 = x \)

We can substitute \( y \) from Equation 1 into Equation 2.

Substituting \( y \) in Equation 2:
\[ 2(2x - 45) - 21 = x \]
Expanding this:
\[ 4x - 90 - 21 = x \]
Combining like terms:
\[ 4x - 111 = x \]
Now, isolate \( x \):
\[ 4x - x = 111 \]
\[ 3x = 111 \]
Dividing both sides by 3:
\[ x = 37 \]

### Step 3: Find \( y \)

Now that we have \( x \), we can find \( y \) using Equation 1:
\[ y = 2x - 45 \]
Substituting \( x = 37 \):
\[ y = 2(37) - 45 \]
Calculating:
\[ y = 74 - 45 \]
\[ y = 29 \]

### Step 4: Conclusion

The two numbers are:
- Greater number \( x = 37 \)
- Smaller number \( y = 29 \)

### Final Answer:
The numbers are 37 and 29.

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