Answer :
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]
[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]