Questions: What is the solution to the system of linear equations represented by the matrix equations below? [ [2 4] [1 2] ] [ [x] [y] ] = [ [6] [3] ] x=3, y=0 x=0, y=3/2 The system of equations has no solution. The system of equations has infinite solutions.

What is the solution to the system of linear equations represented by the matrix equations below?

[ [2 4] [1 2] ] [ [x] [y] ] = [ [6] [3] ]

x=3, y=0
x=0, y=3/2
The system of equations has no solution.
The system of equations has infinite solutions.

Transcript text: What is the solution to the system of linear equations represented by the matrix equations below? \[ \left[\begin{array}{ll} 2 & 4 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 6 \\ 3 \end{array}\right] \] $x=3, y=0$ $x=0, y=\frac{3}{2}$ The system of equations has no solution. The system of equations has infinite solutions.

Solution

Solution Steps

Step 1: Write the System of Equations

The given matrix equation is:

\[ \left[\begin{array}{ll} 2 & 4 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{l} 6 \\ 3 \end{array}\right] \]

This can be translated into the following system of linear equations:

$2x + 4y = 6$
$x + 2y = 3$

Step 2: Solve the System of Equations

We will use the method of substitution or elimination to solve the system. Here, we will use elimination:

Multiply the second equation by 2 to align the coefficients of $x$:

\[ 2(x + 2y) = 2(3) \implies 2x + 4y = 6 \]

Now, the system of equations is:

$2x + 4y = 6$
$2x + 4y = 6$

Step 3: Analyze the System

Both equations are identical, which means they represent the same line. Therefore, the system has infinitely many solutions, as every point on the line is a solution.