Problem 6 - Algebra 2

Question

Use a matrix equation to solve each system of equations. \(3 x+6 y=11\) \(2 x+4 y=7\)

Step-by-Step Solution

Answer

The system has no solutions because the matrix is non-invertible and the simultaneous equations are inconsistent.

Step 1: Write the System in Matrix Form

First, express the system of equations in matrix form. For our system,\[ \begin{align*} 3x + 6y &= 11 \ 2x + 4y &= 7 \end{align*} \]this can be written in matrix form as:\[ \begin{bmatrix} 3 & 6 \ 2 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 11 \ 7 \end{bmatrix} \]

Step 2: Identify Matrices

Identify the matrices: coefficient matrix \(A = \begin{bmatrix} 3 & 6 \ 2 & 4 \end{bmatrix}\), variable matrix \(X = \begin{bmatrix} x \ y \end{bmatrix}\), and constant matrix \(B = \begin{bmatrix} 11 \ 7 \end{bmatrix}\).

Step 3: Check Matrix Invertibility

Check if the matrix \(A\) is invertible by calculating its determinant. The determinant of \(A\) is computed as:\[ \text{det}(A) = (3)(4) - (2)(6) = 12 - 12 = 0 \]Since the determinant is 0, \(A\) is not invertible.

Step 4: Determine if the System Has a Unique Solution

Since the determinant is 0, the matrix \(A\) does not have an inverse, indicating that the system might not have a unique solution. We need to analyze further to determine if there are no solutions or infinitely many solutions. Checking the equations:\(3x + 6y = 11\) is equivalent to \(2x + 4y = 7\) only if both are multiples of each other, but since multiplying the second equation by \(\frac{3}{2}\) yields a different constant term, it's inconsistent, meaning no solutions.

Key Concepts

System of Equations

A system of equations is a set of equations with multiple variables that we aim to solve collectively. In the problem we're discussing, there are two equations with two variables, which are often written like this:

• First equation: \( 3x + 6y = 11 \)
• Second equation: \( 2x + 4y = 7 \)

The goal is to find values for \( x \) and \( y \) that satisfy both equations simultaneously. When solving such systems, we often place them into matrix form for a more structured solution approach. This involves creating a coefficient matrix, a variable matrix, and a constant matrix, facilitating easier application of algebraic methods to find the solution.

Determinant

The determinant is a special number that can be calculated from a square matrix. It gives us important information about the matrix and the system of equations it represents. For a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is determined using the formula:\[\text{det}(A) = (a)(d) - (b)(c)\]In our system, calculating the determinant of the coefficient matrix \( A = \begin{bmatrix} 3 & 6 \ 2 & 4 \end{bmatrix} \) results in:\[\text{det}(A) = (3)(4) - (2)(6) = 12 - 12 = 0\]A determinant of zero indicates that the matrix is singular, meaning it doesn't have an inverse. This often tells us something about the system, such as having no unique solutions or potentially having infinite solutions.

Matrix Invertibility

Matrix invertibility is a key property when solving systems using matrices. If a matrix is invertible (also called nonsingular), it has an inverse that can be used to solve the system uniquely. The inverse of a matrix basically undoes the effect of the matrix, allowing us to isolate the variable matrix.

• For a two-variable system, the coefficient matrix needs a non-zero determinant for it to be invertible.
• If the determinant is zero, it suggests the matrix does not have an inverse, indicating issues with solving the system uniquely via matrix inversion.

In the case of our system, the determinant is zero, so matrix \( A \) is not invertible. This means we cannot directly use matrix inversion to find a unique solution for this system.

Unique Solution

A system of equations will have a unique solution if each variable specifically aligns with one value. This occurs when the corresponding matrix is invertible, which is tied to having a non-zero determinant.

• If the determinant is non-zero, the system has a unique solution.
• If the determinant is zero, it implies the system might have no solution or infinitely many solutions.

For the system \( 3x + 6y = 11 \) and \( 2x + 4y = 7 \), the zero determinant and a pair of equations that don't fully align when factored indicate that the system lacks a unique solution. In our case, further inspection reveals inconsistencies, showing that there are, in fact, no solutions to this set of equations.