Given [tex]\( z = \frac{3}{2} \)[/tex], we need to determine which of the following equations are satisfied:
(a) [tex]\( \frac{3z}{2} + 4 = \frac{25}{4} \)[/tex]
(b) [tex]\( -\frac{3z}{2} + 8 = \frac{41}{4} \)[/tex]
(c) [tex]\( 6z + 15 = 21 \)[/tex]
(d) [tex]\( 12z - 5 = 4 \)[/tex]
Let's check each equation step-by-step:
### Equation (a)
[tex]\[
\frac{3z}{2} + 4 = \frac{25}{4}
\][/tex]
Substitute [tex]\( z = \frac{3}{2} \)[/tex]:
[tex]\[
\frac{3 \left(\frac{3}{2}\right)}{2} + 4 = \frac{25}{4}
\][/tex]
Simplify the left-hand side:
[tex]\[
\frac{3 \cdot \frac{3}{2}}{2} = \frac{3 \cdot 3}{2 \cdot 2} = \frac{9}{4}
\][/tex]
So,
[tex]\[
\frac{9}{4} + 4 = \frac{25}{4}
\][/tex]
Convert 4 to a fraction with denominator 4:
[tex]\[
4 = \frac{16}{4}
\][/tex]
Therefore,
[tex]\[
\frac{9}{4} + \frac{16}{4} = \frac{25}{4}
\][/tex]
This simplifies to:
[tex]\[
\frac{25}{4} = \frac{25}{4}
\][/tex]
Therefore, equation (a) is satisfied.
### Equation (b)
[tex]\[
-\frac{3z}{2} + 8 = \frac{41}{4}
\][/tex]
Substitute [tex]\( z = \frac{3}{2} \)[/tex]:
[tex]\[
-\frac{3 \left(\frac{3}{2}\right)}{2} + 8 = \frac{41}{4}
\][/tex]
Simplify the left-hand side:
[tex]\[
-\frac{3 \cdot \frac{3}{2}}{2} = -\frac{9}{4}
\][/tex]
So,
[tex]\[
-\frac{9}{4} + 8 = \frac{41}{4}
\][/tex]
Convert 8 to a fraction with denominator 4:
[tex]\[
8 = \frac{32}{4}
\][/tex]
Therefore,
[tex]\[
-\frac{9}{4} + \frac{32}{4} = \frac{23}{4}
\][/tex]
This simplifies to:
[tex]\[
\frac{23}{4} \ne \frac{41}{4}
\][/tex]
Therefore, equation (b) is not satisfied.
### Equation (c)
[tex]\[
6z + 15 = 21
\][/tex]
Substitute [tex]\( z = \frac{3}{2} \)[/tex]:
[tex]\[
6 \left(\frac{3}{2}\right) + 15 = 21
\][/tex]
Simplify the left-hand side:
[tex]\[
6 \cdot \frac{3}{2} = 9
\][/tex]
So,
[tex]\[
9 + 15 = 21
\][/tex]
This simplifies to:
[tex]\[
24 \ne 21
\][/tex]
Therefore, equation (c) is not satisfied.
### Equation (d)
[tex]\[
12z - 5 = 4
\][/tex]
Substitute [tex]\( z = \frac{3}{2} \)[/tex]:
[tex]\[
12 \left(\frac{3}{2}\right) - 5 = 4
\][/tex]
Simplify the left-hand side:
[tex]\[
12 \cdot \frac{3}{2} = 18
\][/tex]
So,
[tex]\[
18 - 5 = 4
\][/tex]
This simplifies to:
[tex]\[
13 \ne 4
\][/tex]
Therefore, equation (d) is not satisfied.
### Conclusion
The only equation that is satisfied by [tex]\( z = \frac{3}{2} \)[/tex] is:
[tex]\[
(a) \frac{3z}{2} + 4 = \frac{25}{4}
\][/tex]
Thus, the correct answer is [tex]\( (a) \)[/tex].
