To find the value of [tex]\( x \)[/tex] such that twice the number 37 in base [tex]\( x \)[/tex] is equal to 75 in base [tex]\( x \)[/tex], follow these steps:
1. Interpret the Numbers in their Respective Base [tex]\( x \)[/tex]:
- The number 37 in base [tex]\( x \)[/tex] can be written as:
[tex]\[
3x + 7
\][/tex]
- The number 75 in base [tex]\( x \)[/tex] can be written as:
[tex]\[
7x + 5
\][/tex]
2. Set up the Equation:
- Twice the number 37 in base [tex]\( x \)[/tex] should be equal to 75 in base [tex]\( x \)[/tex]. This gives us:
[tex]\[
2 (3x + 7) = 7x + 5
\][/tex]
3. Expand and Simplify the Equation:
- Distribute the 2 on the left-hand side:
[tex]\[
2(3x + 7) = 6x + 14
\][/tex]
- Now, equate this to the right-hand side:
[tex]\[
6x + 14 = 7x + 5
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[
14 = x + 5
\][/tex]
- Subtract 5 from both sides:
[tex]\[
14 - 5 = x
\][/tex]
[tex]\[
x = 9
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 9 \)[/tex].
