Sure, let's break down the solution step by step.
### Step 1: Convert the Binary Numbers to Decimal
Given the binary equation:
[tex]\[ X_{\text{two}} - 11101_{\text{two}} = 1110100_{\text{two}} \][/tex]
First, we need to convert the known binary numbers to their decimal equivalents.
- For [tex]\(11101_{\text{two}}\)[/tex], to convert it to decimal:
[tex]\[
11101_{\text{two}} = 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
= 16 + 8 + 4 + 0 + 1
= 29_{\text{ten}}
\][/tex]
- For [tex]\(1110100_{\text{two}}\)[/tex], to convert it to decimal:
[tex]\[
1110100_{\text{two}} = 1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
= 64 + 32 + 16 + 0 + 4 + 0 + 0
= 116_{\text{ten}}
\][/tex]
### Step 2: Set Up the Equation in Decimal Form
The equation [tex]\(X_{\text{two}} - 11101_{\text{two}} = 1110100_{\text{two}}\)[/tex] can be rewritten in decimal form using the converted values:
[tex]\[
X - 29 = 116
\][/tex]
### Step 3: Solve for [tex]\(X\)[/tex] in Decimal Form
To find [tex]\(X\)[/tex], we rearrange the equation:
[tex]\[
X = 116 + 29 = 145
\][/tex]
So, [tex]\(X_{\text{ten}} = 145\)[/tex].
### Step 4: Convert [tex]\(X\)[/tex] Back to Binary
Finally, we convert the decimal result [tex]\(145_{\text{ten}}\)[/tex] back to binary:
[tex]\[
145_{\text{ten}} = 1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
= 10010001_{\text{two}}
\][/tex]
So, the binary form of [tex]\(X\)[/tex] is:
[tex]\[
X_{\text{two}} = 10010001_{\text{two}}
\][/tex]
### Conclusion:
Thus, the solution to the equation [tex]\(X_{\text{two}} - 11101_{\text{two}} = 1110100_{\text{two}}\)[/tex] is:
[tex]\[
X = 10010001_{\text{two}}
\][/tex]
