To solve the system of equations for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\begin{aligned}
a + b + c &= \frac{1}{2}, \\
a + b - c &= 6,
\end{aligned}
\][/tex]
we will follow these steps:
1. Label the equations for reference:
[tex]\[
\begin{aligned}
&\text{(1)} \quad a + b + c = \frac{1}{2}, \\
&\text{(2)} \quad a + b - c = 6.
\end{aligned}
\][/tex]
2. Add Equation (1) and Equation (2):
By adding these two equations, we aim to eliminate [tex]\( c \)[/tex].
[tex]\[
\begin{aligned}
&(a + b + c) + (a + b - c) &= \frac{1}{2} + 6 \\
&a + b + c + a + b - c &= \frac{1}{2} + 6 \\
&2a + 2b &= \frac{1}{2} + 6 \\
&2a + 2b &= 6.5 \\
&a + b &= 3.25. \quad \text{(3)}
\end{aligned}
\][/tex]
Equation (3) tells us that [tex]\( a + b = 3.25 \)[/tex].
3. Express [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
From Equation (3), we can write:
[tex]\[
a = 3.25 - b. \quad \text{(4)}
\][/tex]
4. Substitute the expression for [tex]\( a \)[/tex] into one of the original equations to find [tex]\( c \)[/tex]:
Use Equation (1):
[tex]\[
\begin{aligned}
(3.25 - b) + b + c &= \frac{1}{2} \\
3.25 + c &= \frac{1}{2} \\
c &= \frac{1}{2} - 3.25 \\
c &= -2.75.
\end{aligned}
\][/tex]
Thus, we find that [tex]\( c = -2.75 \)[/tex].
Therefore, the final values are:
[tex]\[
a = 3.25 - b \quad \text{and} \quad c = -2.75.
\][/tex]
So, [tex]\( a \)[/tex] is expressed in terms of [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is a constant value [tex]\( -2.75 \)[/tex].
