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The augmented matrix is in row-echelon form. Assume that the variables are [tex]$x$[/tex] and [tex]$y$[/tex] and use back substitution to obtain the solution of the associated system of linear equations.

[tex]\[
\left[\begin{array}{rr|r}
1 & -5 & -7 \\
0 & 1 & -3
\end{array}\right]
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. There is one solution. The solution set is \(\{\square\}\).
(Simplify your answer. Type an ordered pair, using integers or fractions.)

B. There are infinitely many solutions. The solution set is the set of all ordered pairs \(\{(\square, y)\}\), where \(y\) is any \(\square\) real number.
(Type an expression using \(y\) as the variable. Simplify your answer.)

C. The system is inconsistent. The solution set is [tex]\(\varnothing\)[/tex].



Answer :

GinnyAnswer
To solve the given system of linear equations using the augmented matrix in row-echelon form:

[tex]\[ \left[\begin{array}{rr|r} 1 & -5 & -7 \\ 0 & 1 & -3 \end{array}\right] \][/tex]

we will use back substitution.

1. Interpret the matrix as a system of equations:
- The first row translates to \( 1x - 5y = -7 \).
- The second row translates to \( 0x + 1y = -3 \), which simplifies to \( y = -3 \).

2. Solve for \( y \) first:
- From the second row, we have \( y = -3 \).

3. Substitute \( y = -3 \) into the first equation to solve for \( x \):
- The first equation is \( x - 5y = -7 \).
- Substitute \( y = -3 \) into this equation:
[tex]\[ x - 5(-3) = -7 \][/tex]
- Simplify the equation:
[tex]\[ x + 15 = -7 \][/tex]
- Solve for \( x \):
[tex]\[ x = -7 - 15 \][/tex]
[tex]\[ x = -22 \][/tex]

Therefore, the solution to the system of linear equations is the ordered pair \((-22, -3)\).

Select the correct choice:
A. There is one solution. The solution set is [tex]\(\{(-22, -3)\}\)[/tex].