To solve the system of linear equations:
[tex]\[
\begin{aligned}
4a - 24b &= -13, \\
8a + 45c &= 41, \\
48b + 35c &= 61,
\end{aligned}
\][/tex]
we can proceed with the following steps:
1. Write the equations in matrix form:
The given system of equations can be represented in matrix form as [tex]\( Ax = B \)[/tex], where [tex]\( A \)[/tex] is the coefficient matrix, [tex]\( x \)[/tex] is the vector of variables, and [tex]\( B \)[/tex] is the constant vector.
[tex]\[
\begin{pmatrix}
4 & -24 & 0 \\
8 & 0 & 45 \\
0 & 48 & 35
\end{pmatrix}
\begin{pmatrix}
a \\
b \\
c
\end{pmatrix}
=
\begin{pmatrix}
-13 \\
41 \\
61
\end{pmatrix}
\][/tex]
2. Solve for the variables:
Using matrix methods or substitution and elimination, we can solve this system of equations.
3. Find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
After performing the calculations, we obtain the following solutions for the variables:
[tex]\[
\begin{aligned}
a &= 1.75, \\
b &= 0.8333333333333334, \\
c &= 0.6.
\end{aligned}
\][/tex]
Thus, the solutions to the system of equations are:
[tex]\[
\boxed{a = 1.75, \; b = 0.8333333333333334, \; c = 0.6}
\][/tex]
