The equations \(x - 2y = 6{\text{ and }}3x - 6y = 18\) has infinitely many solutions. Option (D) is correct.
Further explanation:
Consider \({a_1}x + {b_1}y + {c_1}\) and \({a_2}x + {b_2}y + {c_2}.\)
If \(\dfrac{{{a_1}}}{{{a_2}}}
e \dfrac{{{b_1}}}{{{b_2}}}\) then the system of equation has exactly one solution and the system of equations are consistent.
If \(\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\) then the system of equation has infinite many solution and the system of equations are consistent.
If \(\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}}
e \dfrac{{{c_1}}}{{{c_2}}}\) then the system of equation has no solution and the system of equations are inconsistent.
Given:
The options are as follows,
A. \(\left( {2, - 2} \right)\)
B. \(\left( {3,{\text{negative three halves}}} \right)\)
C. No Solutions
D. Infinitely Many Solutions
Explanation:
The equations are \(x - 2y = 6{\text{ and }}3x - 6y = 18.\)
\({a_1} = 1,{b_1} = - 2{\text{ and }}{c_1} = 6\)
\({a_2} = 3,{b_2} = - 6{\text{ and }}{c_2} = 18\)
The ratio of \(\dfrac{{{a_1}}}{{{a_2}}}, \dfrac{{{b_1}}}{{{b_2}}}\, \text{and}\, \dfrac{{{c_1}}}{{{c_2}}}\) can be calculated as follows,
\(\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{3}\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{3}\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{1}{3}\)
The system of equations has infinite many solutions.
The equations \(x - 2y = 6{\text{ and }}3x - 6y = 18\) has infinitely many solutions. Option (D) is correct.
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Linear equation
Keywords: consistent, inconsistent, equations, system of equations, parallel lines, intersecting lines, coincident lines, no solution, many solutions, one solution.
