To determine the value of [tex]\(abc\)[/tex] given the equations [tex]\(a + b = 5\)[/tex], [tex]\(b + c = 7\)[/tex], and [tex]\(c + a = 6\)[/tex], follow these steps:
1. Label the equations:
[tex]\[
\begin{cases}
a + b = 5 & \text{(1)} \\
b + c = 7 & \text{(2)} \\
c + a = 6 & \text{(3)}
\end{cases}
\][/tex]
2. Add all three equations:
[tex]\[
(a + b) + (b + c) + (c + a) = 5 + 7 + 6
\][/tex]
Simplify this to:
[tex]\[
2a + 2b + 2c = 18
\][/tex]
Factor out the 2:
[tex]\[
a + b + c = 9 \quad \text{(4)}
\][/tex]
3. Use equation (4) to find individual values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- From equation (1):
[tex]\[
a + b = 5
\][/tex]
Subtract equation (1) from equation (4):
[tex]\[
c = (a + b + c) - (a + b) = 9 - 5 = 4
\][/tex]
- From equation (2):
[tex]\[
b + c = 7
\][/tex]
Subtract equation (3) from equation (4):
[tex]\[
b = (a + b + c) - (c + a) = 9 - 6 = 3
\][/tex]
- From equation (3):
[tex]\[
c + a = 6
\][/tex]
Subtract equation (2) from equation (4):
[tex]\[
a = (a + b + c) - (b + c) = 9 - 7 = 2
\][/tex]
4. Substitute these values to verify:
- For [tex]\(a\)[/tex]:
[tex]\[
a = 2
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
b = 3
\][/tex]
- For [tex]\(c\)[/tex]:
[tex]\[
c = 4
\][/tex]
5. Calculate the value of [tex]\(abc\)[/tex]:
[tex]\[
abc = 2 \times 3 \times 4 = 24
\][/tex]
Hence, the value of [tex]\(abc\)[/tex] is [tex]\(24\)[/tex].
Therefore, the correct answer is:
b. 24
