IDNLearn.com provides a comprehensive platform for finding accurate answers. Our community provides timely and precise responses to help you understand and solve any issue you face.
Sagot :
Certainly! To solve the given system of linear equations using inverse matrices, let's follow the detailed step-by-step solution:
### Step 1: Write the system of equations in matrix form
The system of linear equations given is:
[tex]\[ \begin{cases} 5x - 2y = 30 \\ x + 6y = -26 \end{cases} \][/tex]
We can represent this system in matrix form as [tex]\(Ax = B\)[/tex], where:
[tex]\[ A = \begin{pmatrix} 5 & -2 \\ 1 & 6 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 30 \\ -26 \end{pmatrix} \][/tex]
Here [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(x\)[/tex] is the vector of variables, and [tex]\(B\)[/tex] is the constant matrix.
### Step 2: Find the inverse of the coefficient matrix [tex]\(A\)[/tex]
The inverse of matrix [tex]\(A\)[/tex], denoted as [tex]\(A^{-1}\)[/tex], is needed to solve for [tex]\(x\)[/tex]. Given:
[tex]\[ A = \begin{pmatrix} 5 & -2 \\ 1 & 6 \end{pmatrix} \][/tex]
The inverse matrix [tex]\(A^{-1}\)[/tex] can be represented as:
[tex]\[ A^{-1} = \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \][/tex]
### Step 3: Multiply [tex]\(A^{-1}\)[/tex] by [tex]\(B\)[/tex] to find [tex]\(x\)[/tex]
To find the solution vector [tex]\(x\)[/tex], we use the relationship:
[tex]\[ x = A^{-1}B \][/tex]
Now, let's perform the matrix multiplication:
[tex]\[ x = \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \begin{pmatrix} 30 \\ -26 \end{pmatrix} \][/tex]
### Step 4: Calculate the resulting vector
Perform the multiplication:
[tex]\[ \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \begin{pmatrix} 30 \\ -26 \end{pmatrix} = \begin{pmatrix} (0.1875 \cdot 30) + (0.0625 \cdot -26) \\ (-0.03125 \cdot 30) + (0.15625 \cdot -26) \end{pmatrix} \][/tex]
Calculate each component:
[tex]\[ \begin{pmatrix} (0.1875 \cdot 30) + (0.0625 \cdot -26) \\ (-0.03125 \cdot 30) + (0.15625 \cdot -26) \end{pmatrix} = \begin{pmatrix} 5.625 - 1.625 \\ -0.9375 - 4.0625 \end{pmatrix} = \begin{pmatrix} 4 \\ -5 \end{pmatrix} \][/tex]
### Step 5: Interpret the results
Therefore, the solution to the system of equations is:
[tex]\[ \begin{cases} x = 4 \\ y = -5 \end{cases} \][/tex]
So, the solution is:
[tex]\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ -5 \end{pmatrix} \][/tex]
This represents the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations in the given system.
### Step 1: Write the system of equations in matrix form
The system of linear equations given is:
[tex]\[ \begin{cases} 5x - 2y = 30 \\ x + 6y = -26 \end{cases} \][/tex]
We can represent this system in matrix form as [tex]\(Ax = B\)[/tex], where:
[tex]\[ A = \begin{pmatrix} 5 & -2 \\ 1 & 6 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 30 \\ -26 \end{pmatrix} \][/tex]
Here [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(x\)[/tex] is the vector of variables, and [tex]\(B\)[/tex] is the constant matrix.
### Step 2: Find the inverse of the coefficient matrix [tex]\(A\)[/tex]
The inverse of matrix [tex]\(A\)[/tex], denoted as [tex]\(A^{-1}\)[/tex], is needed to solve for [tex]\(x\)[/tex]. Given:
[tex]\[ A = \begin{pmatrix} 5 & -2 \\ 1 & 6 \end{pmatrix} \][/tex]
The inverse matrix [tex]\(A^{-1}\)[/tex] can be represented as:
[tex]\[ A^{-1} = \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \][/tex]
### Step 3: Multiply [tex]\(A^{-1}\)[/tex] by [tex]\(B\)[/tex] to find [tex]\(x\)[/tex]
To find the solution vector [tex]\(x\)[/tex], we use the relationship:
[tex]\[ x = A^{-1}B \][/tex]
Now, let's perform the matrix multiplication:
[tex]\[ x = \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \begin{pmatrix} 30 \\ -26 \end{pmatrix} \][/tex]
### Step 4: Calculate the resulting vector
Perform the multiplication:
[tex]\[ \begin{pmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{pmatrix} \begin{pmatrix} 30 \\ -26 \end{pmatrix} = \begin{pmatrix} (0.1875 \cdot 30) + (0.0625 \cdot -26) \\ (-0.03125 \cdot 30) + (0.15625 \cdot -26) \end{pmatrix} \][/tex]
Calculate each component:
[tex]\[ \begin{pmatrix} (0.1875 \cdot 30) + (0.0625 \cdot -26) \\ (-0.03125 \cdot 30) + (0.15625 \cdot -26) \end{pmatrix} = \begin{pmatrix} 5.625 - 1.625 \\ -0.9375 - 4.0625 \end{pmatrix} = \begin{pmatrix} 4 \\ -5 \end{pmatrix} \][/tex]
### Step 5: Interpret the results
Therefore, the solution to the system of equations is:
[tex]\[ \begin{cases} x = 4 \\ y = -5 \end{cases} \][/tex]
So, the solution is:
[tex]\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ -5 \end{pmatrix} \][/tex]
This represents the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations in the given system.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.
