Question:
The system of equations $x+y+2z=4$, $3x+3y+6z=17$, $5x-3y+2z=27$ has
The system of equations $x+y+2z=4$, $3x+3y+6z=17$, $5x-3y+2z=27$ has
Show Hint
Logic Tip: Before calculating large determinants using Cramer's Rule or performing full Gaussian elimination, always quickly scan the equations to see if one row is a scalar multiple of another! It saves massive amounts of time.
Updated On: Apr 30, 2026
- no solution
- finitely many solutions
- infinitely many solutions
- unique and trivial solution
- unique and non-trivial solution
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
A system of linear equations in three variables geometrically represents three planes. If two of the planes are parallel but not identical (meaning their variable coefficients are proportional, but their constant terms are not), they will never intersect. In this case, the entire system is inconsistent and has no solution.
Step 1: Write down the given system of equations.
Equation 1: $x + y + 2z = 4$ Equation 2: $3x + 3y + 6z = 17$ Equation 3: $5x - 3y + 2z = 27$
Step 2: Inspect the variable coefficients for proportionality.
Compare the coefficients of the variables $(x, y, z)$ in Equation 1 and Equation 2: Equation 1 coefficients: $(1, 1, 2)$ Equation 2 coefficients: $(3, 3, 6)$ Notice that multiplying the coefficients of Equation 1 by $3$ gives the exact coefficients of Equation 2.
Step 3: Multiply the entire first equation by 3.
To see if the planes are identical or parallel, multiply Equation 1 by the scalar $3$: $$3(x + y + 2z) = 3(4)$$ $$3x + 3y + 6z = 12$$
Step 4: Compare the manipulated equation with Equation 2.
Let's put the manipulated Equation 1 right next to the original Equation 2: Manipulated Eq 1: $3x + 3y + 6z = 12$ Original Eq 2: $3x + 3y + 6z = 17$
Step 5: Analyze the contradiction and conclude.
The left sides of both equations are identical ($3x + 3y + 6z$), but the right sides are different ($12$ vs $17$). It is impossible for the same quantity to simultaneously equal 12 and 17. Geometrically, this means the two planes are strictly parallel and never intersect. Because all three planes must intersect at a common point for a solution to exist, the system is inconsistent. Hence the correct answer is (A) no solution.
A system of linear equations in three variables geometrically represents three planes. If two of the planes are parallel but not identical (meaning their variable coefficients are proportional, but their constant terms are not), they will never intersect. In this case, the entire system is inconsistent and has no solution.
Step 1: Write down the given system of equations.
Equation 1: $x + y + 2z = 4$ Equation 2: $3x + 3y + 6z = 17$ Equation 3: $5x - 3y + 2z = 27$
Step 2: Inspect the variable coefficients for proportionality.
Compare the coefficients of the variables $(x, y, z)$ in Equation 1 and Equation 2: Equation 1 coefficients: $(1, 1, 2)$ Equation 2 coefficients: $(3, 3, 6)$ Notice that multiplying the coefficients of Equation 1 by $3$ gives the exact coefficients of Equation 2.
Step 3: Multiply the entire first equation by 3.
To see if the planes are identical or parallel, multiply Equation 1 by the scalar $3$: $$3(x + y + 2z) = 3(4)$$ $$3x + 3y + 6z = 12$$
Step 4: Compare the manipulated equation with Equation 2.
Let's put the manipulated Equation 1 right next to the original Equation 2: Manipulated Eq 1: $3x + 3y + 6z = 12$ Original Eq 2: $3x + 3y + 6z = 17$
Step 5: Analyze the contradiction and conclude.
The left sides of both equations are identical ($3x + 3y + 6z$), but the right sides are different ($12$ vs $17$). It is impossible for the same quantity to simultaneously equal 12 and 17. Geometrically, this means the two planes are strictly parallel and never intersect. Because all three planes must intersect at a common point for a solution to exist, the system is inconsistent. Hence the correct answer is (A) no solution.
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM System of Linear Equations Questions
- The following system of equations \[ x + y + z = 1 \] \[ 2x + 3y - mz = 2 \] \[ 3x + 5y + 3z = 3 \] has no unique solution. Then the value of \( m \) is equal to
- If the following system of linear equations
\[ x-2y+z=5 \] \[ 2x-y+2z=7 \] \[ x+2y+\lambda z=5 \] has a unique solution, then \(\lambda \ne\)
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions