Sure, let's solve this step by step.
We are given two conditions involving two numbers. Let’s denote the first number by [tex]\( x \)[/tex] and the second number by [tex]\( y \)[/tex].
### Step 1: Formulating the Equations
From the problem, we can derive the following equations:
1. If 11 is added to the first number, the result is twice the second number:
[tex]\[ x + 11 = 2y \][/tex]
2. If 20 is added to the second number, the result is twice the first number:
[tex]\[ y + 20 = 2x \][/tex]
### Step 2: Solving the System of Equations
We have the system of linear equations:
[tex]\[
\begin{cases}
x + 11 = 2y \\
y + 20 = 2x
\end{cases}
\][/tex]
First, let's solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x + 11 = 2y \][/tex]
[tex]\[ x = 2y - 11 \][/tex]
Now we substitute this value of [tex]\( x \)[/tex] into the second equation:
[tex]\[ y + 20 = 2(2y - 11) \][/tex]
[tex]\[ y + 20 = 4y - 22 \][/tex]
Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ y + 20 = 4y - 22 \][/tex]
[tex]\[ 20 + 22 = 4y - y \][/tex]
[tex]\[ 42 = 3y \][/tex]
[tex]\[ y = 14 \][/tex]
Now that we have [tex]\( y \)[/tex], substitute it back into the expression for [tex]\( x \)[/tex]:
[tex]\[ x = 2y - 11 \][/tex]
[tex]\[ x = 2(14) - 11 \][/tex]
[tex]\[ x = 28 - 11 \][/tex]
[tex]\[ x = 17 \][/tex]
### Step 3: Verifying the Solution
To ensure our solutions [tex]\( x = 17 \)[/tex] and [tex]\( y = 14 \)[/tex] are correct, we can substitute them back into the original equations:
For the first equation:
[tex]\[ x + 11 = 2y \][/tex]
[tex]\[ 17 + 11 = 2 \times 14 \][/tex]
[tex]\[ 28 = 28 \][/tex] (True)
For the second equation:
[tex]\[ y + 20 = 2x \][/tex]
[tex]\[ 14 + 20 = 2 \times 17 \][/tex]
[tex]\[ 34 = 34 \][/tex] (True)
Both equations are satisfied.
### Conclusion
Thus, the numbers are [tex]\( \mathbf{17} \)[/tex] and [tex]\( \mathbf{14} \)[/tex].
