Answer :
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].
[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].