Math Problem Statement
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
−2x1
−
3x2
=
−15
6x1
+
x2
=
−27 -2 -3 6 1
x1
x2
=
−15
−27
x1
x2
= -6 9
Solution
To express the given system of linear equations in matrix form A x = b, we need to rewrite the equations and solve for x using matrix operations.
Step 1: Write the system of equations.
The given system of equations is:
-2x_1 - 3x_2 &= -15 \\ 6x_1 + x_2 &= -27 \end{aligned}$$ ### Step 2: Express in matrix form **A** **x** = **b**. We can represent the system as a matrix equation: $$\begin{pmatrix} -2 & -3 \\ 6 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} -15 \\ -27 \end{pmatrix}$$ This is in the form **A** **x** = **b**, where: $$\mathbf{A} = \begin{pmatrix} -2 & -3 \\ 6 & 1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -15 \\ -27 \end{pmatrix}$$ ### Step 3: Solve for **x**. To solve for **x**, we will find the inverse of matrix **A** (if it exists) and multiply it by **b**: $$\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}$$ Let's calculate this step. I'll now compute the inverse of matrix **A** and solve the system.The solution to the system of equations is: $$\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} -6 \\ 9 \end{pmatrix}$$ Thus, $$x_1 = -6$$ and $$x_2 = 9$$. Would you like more details or have any questions? Here are some related questions you might find helpful: 1. How do you calculate the inverse of a 2x2 matrix manually? 2. How can you check if a matrix is invertible? 3. What happens if the determinant of a matrix is zero? 4. How does matrix multiplication work in solving systems of equations? 5. What other methods can be used to solve systems of linear equations? **Tip:** Always check if a matrix is invertible by calculating its determinant before attempting to solve using the inverse method!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Formulas
Matrix form: A * x = b
Inverse of a matrix: A^(-1)
Solution: x = A^(-1) * b
Theorems
Matrix Inverse Theorem
Determinant and Invertibility
Suitable Grade Level
Grades 10-12
Related Recommendation
Solve System of Linear Equations Using Matrix Inversion
Solving a System of Linear Equations Using Inverse Matrices: 4x1+x2-5x3=16, x1 +2x2-4x3=6, -2x2-x3=6
Solving a System of Linear Equations using Matrices and Inverse Matrix Method
Solve a System of Linear Equations using Matrix Inversion
Solving Linear Systems with Matrix Method