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Answer :
Sure, let's go through a detailed, step-by-step solution for each of the given algebra problems.
1. Find the value of [tex]\( x \)[/tex] for which [tex]\( 16_x + 32_x = 40_{\text{ten}} \)[/tex]:
Start by expressing the numbers [tex]\( 16 \)[/tex] and [tex]\( 32 \)[/tex] in base [tex]\( x \)[/tex]:
- [tex]\( 16_x \)[/tex] is [tex]\( 1 \cdot x + 6 \)[/tex].
- [tex]\( 32_x \)[/tex] is [tex]\( 3 \cdot x + 2 \)[/tex].
The equation becomes:
[tex]\[
(1 \cdot x + 6) + (3 \cdot x + 2) = 40
\][/tex]
Simplifying the left side:
[tex]\[
x + 6 + 3x + 2 = 40
\][/tex]
[tex]\[
4x + 8 = 40
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
4x = 40 - 8
\][/tex]
[tex]\[
4x = 32
\][/tex]
[tex]\[
x = \frac{32}{4} = 8
\][/tex]
Thus, [tex]\( x = 8 \)[/tex].
2. Find the value of [tex]\( x \)[/tex] such that [tex]\( 324_x = 89 \)[/tex]:
Break down [tex]\( 324_x \)[/tex] in terms of [tex]\( x \)[/tex]:
- [tex]\( 324_x \)[/tex] is [tex]\( 3 \cdot x^2 + 2 \cdot x + 4 \)[/tex].
Set up the equation:
[tex]\[
3 \cdot x^2 + 2 \cdot x + 4 = 89
\][/tex]
We can test different base values to find the correct [tex]\( x \)[/tex]. By evaluating:
[tex]\[
3 \cdot 5^2 + 2 \cdot 5 + 4 = 3 \cdot 25 + 10 + 4 = 75 + 10 + 4 = 89
\][/tex]
This calculation satisfies the equation when [tex]\( x = 5 \)[/tex].
Therefore, for the given problems, the correct answers are:
1. [tex]\( x = 8 \)[/tex] for the first equation problem.
2. [tex]\( x = 5 \)[/tex] for the second equation problem.
1. Find the value of [tex]\( x \)[/tex] for which [tex]\( 16_x + 32_x = 40_{\text{ten}} \)[/tex]:
Start by expressing the numbers [tex]\( 16 \)[/tex] and [tex]\( 32 \)[/tex] in base [tex]\( x \)[/tex]:
- [tex]\( 16_x \)[/tex] is [tex]\( 1 \cdot x + 6 \)[/tex].
- [tex]\( 32_x \)[/tex] is [tex]\( 3 \cdot x + 2 \)[/tex].
The equation becomes:
[tex]\[
(1 \cdot x + 6) + (3 \cdot x + 2) = 40
\][/tex]
Simplifying the left side:
[tex]\[
x + 6 + 3x + 2 = 40
\][/tex]
[tex]\[
4x + 8 = 40
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
4x = 40 - 8
\][/tex]
[tex]\[
4x = 32
\][/tex]
[tex]\[
x = \frac{32}{4} = 8
\][/tex]
Thus, [tex]\( x = 8 \)[/tex].
2. Find the value of [tex]\( x \)[/tex] such that [tex]\( 324_x = 89 \)[/tex]:
Break down [tex]\( 324_x \)[/tex] in terms of [tex]\( x \)[/tex]:
- [tex]\( 324_x \)[/tex] is [tex]\( 3 \cdot x^2 + 2 \cdot x + 4 \)[/tex].
Set up the equation:
[tex]\[
3 \cdot x^2 + 2 \cdot x + 4 = 89
\][/tex]
We can test different base values to find the correct [tex]\( x \)[/tex]. By evaluating:
[tex]\[
3 \cdot 5^2 + 2 \cdot 5 + 4 = 3 \cdot 25 + 10 + 4 = 75 + 10 + 4 = 89
\][/tex]
This calculation satisfies the equation when [tex]\( x = 5 \)[/tex].
Therefore, for the given problems, the correct answers are:
1. [tex]\( x = 8 \)[/tex] for the first equation problem.
2. [tex]\( x = 5 \)[/tex] for the second equation problem.
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