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Select the correct systems of equations.

Which systems of linear equations have no solution?

A.
[tex]\[
\begin{array}{l}
x+y+z=1,100 \\
x-2y-z=-500 \\
2x+3y+2z=2,600
\end{array}
\][/tex]

B.
[tex]\[
\begin{array}{l}
x+y+z=1,400 \\
x-2y-z=-500 \\
2x+2y+2z=2,700
\end{array}
\][/tex]

C.
[tex]\[
\begin{array}{l}
x+y+z=1,900 \\
x-y-2z=-2,000 \\
2x+2y+z=1,100
\end{array}
\][/tex]

D.
[tex]\[
\begin{array}{l}
x+y+z=1,500 \\
x-y-z=-500 \\
2x+y+z=2,000
\end{array}
\][/tex]

E.
[tex]\[
\begin{array}{l}
x+y+z=1,400 \\
-0.5x-0.5y-0.5z=-900 \\
2x+3y+2z=3,000
\end{array}
\][/tex]

F.
[tex]\[
\begin{array}{l}
x+y+z=2,400 \\
2x-2y+2z=700 \\
x+3y+z=2,400
\end{array}
\][/tex]



Answer :

GinnyAnswer
To determine which systems of linear equations have no solution, we examine each provided system:

1. The first system:
[tex]\[ \begin{array}{l} x + y + z = 1100 \\ x - 2y - z = -500 \\ 2x + 3y + 2z = 2600 \end{array} \][/tex]

2. The second system:
[tex]\[ \begin{array}{l} x + y + z = 1400 \\ x - 2y - z = -500 \\ 2x + 2y + 2z = 2700 \end{array} \][/tex]

3. The third system:
[tex]\[ \begin{array}{l} x + y + z = 1900 \\ x - y - 2z = -2000 \\ 2x + 2y + z = 1100 \end{array} \][/tex]

4. The fourth system:
[tex]\[ \begin{array}{l} x + y + z = 1500 \\ x - y - z = -500 \\ 2x + y + z = 2000 \end{array} \][/tex]

5. The fifth system:
[tex]\[ \begin{array}{l} x + y + z = 1400 \\ -0.5x - 0.5y - 0.5z = -900 \\ 2x + 3y + 2z = 3000 \end{array} \][/tex]

6. The sixth system:
[tex]\[ \begin{array}{l} x + y + z = 2400 \\ 2x - 2y + 2z = 700 \\ x + 3y + z = 2400 \end{array} \][/tex]

After solving these systems, we find that only systems 2, 5, and 6 have no solution. Therefore, these are the systems without a solution:
[tex]\[ \begin{array}{l} x + y + z = 1400 \\ x - 2y - z = -500 \\ 2x + 2y + 2z = 2700 \end{array} \][/tex]
[tex]\[ \begin{array}{l} x + y + z = 1400 \\ -0.5x - 0.5y - 0.5z = -900 \\ 2x + 3y + 2z = 3000 \end{array} \][/tex]
[tex]\[ \begin{array}{l} x + y + z = 2400 \\ 2x - 2y + 2z = 700 \\ x + 3y + z = 2400 \end{array} \][/tex]

So, the systems of linear equations that have no solution are the second, fifth, and sixth sets of equations.

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