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Solve the following system of equations:

[tex]\[
\begin{aligned}
4a - 24b &= -13 \\
8a + 45c &= 41 \\
48b + 35c &= 61
\end{aligned}
\][/tex]


Sagot :

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]