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Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4
x1 + x2 + x3 + x4 = 4,
x1 + 2x2 + 3x3 + 4x4 = 5,
x1 + 3x2 + 5x3 + kx4 = 5.
If the system has no solutions, then k = _____
x1 + x2 + x3 + x4 = 4,
x1 + 2x2 + 3x3 + 4x4 = 5,
x1 + 3x2 + 5x3 + kx4 = 5.
If the system has no solutions, then k = _____
Correct answer is between '7,7'. Can you explain this answer?
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Most Upvoted Answer
Consider the following system of three linear equations in four unknow...
**Solution:**
To determine the value of k, we need to analyze the given system of equations and check if it has a solution or not.
Let's write the given system of equations in matrix form:
\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
4 \\
5 \\
5
\end{bmatrix}
\]
To determine if the system has a solution, we can perform row operations on the augmented matrix [A | B] and check for row echelon form.
**Step 1: Interchange R1 and R2**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
1 & 1 & 1 & 1 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
4 \\
5
\end{bmatrix}
\]
**Step 2: Subtract R1 from R2**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
5
\end{bmatrix}
\]
**Step 3: Subtract R1 from R3**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
0 & 1 & 2 & k-4
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
1
\end{bmatrix}
\]
**Step 4: Add R2 to R3**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
0 & 0 & 0 & k-1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
0
\end{bmatrix}
\]
From the row echelon form, we can observe that the third row implies 0 = k-1. In order for the system to have a solution, the coefficient of x4 in the third equation must be zero. Therefore, k-1 = 0, which implies k = 1.
However, the question states that the system has no solution. This means that the third equation must be inconsistent, i.e., the coefficient of x4 in the third equation
To determine the value of k, we need to analyze the given system of equations and check if it has a solution or not.
Let's write the given system of equations in matrix form:
\[
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
4 \\
5 \\
5
\end{bmatrix}
\]
To determine if the system has a solution, we can perform row operations on the augmented matrix [A | B] and check for row echelon form.
**Step 1: Interchange R1 and R2**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
1 & 1 & 1 & 1 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
4 \\
5
\end{bmatrix}
\]
**Step 2: Subtract R1 from R2**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
1 & 3 & 5 & k
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
5
\end{bmatrix}
\]
**Step 3: Subtract R1 from R3**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
0 & 1 & 2 & k-4
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
1
\end{bmatrix}
\]
**Step 4: Add R2 to R3**
\[
\begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -2 & -3 \\
0 & 0 & 0 & k-1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-1 \\
0
\end{bmatrix}
\]
From the row echelon form, we can observe that the third row implies 0 = k-1. In order for the system to have a solution, the coefficient of x4 in the third equation must be zero. Therefore, k-1 = 0, which implies k = 1.
However, the question states that the system has no solution. This means that the third equation must be inconsistent, i.e., the coefficient of x4 in the third equation
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Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer?
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Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer?.
Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer?.
Solutions for Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4x1+ x2 + x3 + x4 = 4,x1 + 2x2 + 3x3 + 4x4 = 5,x1 + 3x2 + 5x3 + kx4 = 5.If the system has no solutions, then k =_____Correct answer is between '7,7'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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