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Using the matrix solver on your calculator, find the solution to the system of equations shown below.

[tex]\[
\begin{array}{r}
3x - y = 4 \\
6x - 2y = 7
\end{array}
\][/tex]

A. More than 1 solution
B. [tex]\( x = 6, y = 2 \)[/tex]
C. No solution
D. [tex]\( x = 3, y = 1 \)[/tex]


Sagot :

To solve the system of equations using matrix methods, we first need to put the system into an augmented matrix form. The system of equations is:

[tex]\[ \begin{cases} 3x - y = 4 & (1) \\ 6x - 2y = 7 & (2) \end{cases} \][/tex]

The augmented matrix for this system of equations will be:

[tex]\[ \begin{pmatrix} 3 & -1 & | & 4 \\ 6 & -2 & | & 7 \end{pmatrix} \][/tex]

We will use row operations to attempt to find a solution. Let's make the augmented matrix row-echelon form.

First, observe the relationship between the rows. Notice that the second row is just twice the first row when considering the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:

[tex]\[ \begin{pmatrix} 3 & -1 & | & 4 \\ 6 & -2 & | & 7 \end{pmatrix} \][/tex]

[tex]\[ R2 = 2 \cdot R1 \][/tex]

However, if we double the first row, we get:

[tex]\[ \begin{cases} 6x - 2y = 8 \end{cases} \][/tex]

Comparing these, we see that:

[tex]\[ \begin{pmatrix} 6 & -2 & | & 8 \\ 6 & -2 & | & 7 \end{pmatrix} \][/tex]

Since [tex]\(8 \neq 7\)[/tex], the equations are inconsistent. This means there is no single pair [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.

Thus, the system has no solution.

This leads us to the correct answer:

C. No solution