The equations [tex]x - 2y = 6{\text{ and }}3x - 6y = 18[/tex] has infinitely many solutions. Option (D) is correct.
Further explanation:
Consider [tex]{a_1}x + {b_1}y + {c_1}[/tex] and [tex]{a_2}x + {b_2}y + {c_2}.[/tex]
If [tex]\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}[/tex] then the system of equation has exactly one solution and the system of equations are consistent.
If [tex]\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}[/tex] then the system of equation has infinite many solution and the system of equations are consistent.
If [tex]\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}[/tex] then the system of equation has no solution and the system of equations are inconsistent.
Given:
The options are as follows,
A. [tex]\left( {2, - 2} \right)[/tex]
B. [tex]\left( {3,{\text{negative three halves}}} \right)[/tex]
C. No Solutions
D. Infinitely Many Solutions
Explanation:
The equations are [tex]x - 2y = 6{\text{ and }}3x - 6y = 18.[/tex]
[tex]{a_1} = 1,{b_1} = - 2{\text{ and }}{c_1} = 6[/tex]
[tex]{a_2} = 3,{b_2} = - 6{\text{ and }}{c_2} = 18[/tex]
The ratio of [tex]\dfrac{{{a_1}}}{{{a_2}}}, \dfrac{{{b_1}}}{{{b_2}}}\, \text{and}\, \dfrac{{{c_1}}}{{{c_2}}}[/tex] can be calculated as follows,
[tex]\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{3}\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{3}\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{1}{3}[/tex]
The system of equations has infinite many solutions.
The equations [tex]x - 2y = 6{\text{ and }}3x - 6y = 18[/tex] 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.
