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$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-algebra-multiple-choice-question-211"}],"about":{"@type":"Thing","name":"Math, Algebra, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Algebra","Duplicate"],"name":"

$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

","comment":{"@type":"Comment","text":"

Choice B is correct. A system of two linear equations in two variables, $x$ and $y$, has no solution if the lines represented by the equations in the $xy$-plane are parallel and distinct. Lines represented by equations in standard form, $Ax+By=C$ and $Dx+By=F$, are parallel if the coefficients for $x$ and $y$ in one equation are proportional to the corresponding coefficients in the other equation, meaning $\\frac{D}{A}=\\frac{E}{B}$; and the lines are distinct if the constants are not proportional, meaning $\\frac{F}{C}$ is not equal to $\\frac{D}{A}$ or $\\frac{E}{B}$. The given equation, $y=6x+18$, can be written in standard form by subtracting $6x$ from both sides of the equation to yield $-6x+y=18$. Therefore, the given equation can be written in the form $Ax+By=C$, where $A=-6,B=1$, and $C=18$. The equation in choice B, $-6x+y=22$, is written in the form $Dx+By=F$, where $D=-6,E=1$, and $F=22$. Therefore, $\\frac{D}{A}=\\frac{-6}{-6}$, which can be rewritten as $\\frac{D}{A}=1$; $\\frac{E}{B}=\\frac{1}{1}$, which can be rewritten as $\\frac{E}{B}=1$; and $\\frac{F}{C}=\\frac{22}{18}$, which can be rewritten as $\\frac{F}{C}=\\frac{11}{9}$. Since $\\frac{D}{A}=1$, $\\frac{E}{B}=1$, and $\\frac{F}{C}$ is not equal to 1, it follows that the given equation and the equation $-6x+y=22$ are parallel and distinct. Therefore, a system of two linear equations consisting of the given equation and the equation $-6x+y=22$ has no solution. Thus, the equation in choice B could be the second equation in the system.

Choice A is incorrect. The equation $-6x+y=18$ and the given equation represent the same line in the $xy$-plane. Therefore, a system of these linear equations would have infinitely many solutions, rather than no solution.

Choice C is incorrect. The equation $-12x+y=36$ and the given equation represent lines in the $xy$-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.

Choice D is incorrect. The equation $-12x+y=18$ and the given equation represent lines in the $xy$-plane that are distinct and not parallel. Therefore, a system of these linear equations would have exactly one solution, rather than no solution.

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$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$-6x+y=18$

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$-12x+y=36$

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$-12x+y=18$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$-6x+y=22$

","comment":{"@type":"Comment","text":"

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"}}},"url":"https://classnotes.xyz/mcqs/sat-math-algebra-multiple-choice-question-211"}

$$y=6x+18$$One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

$$y=6x+18$$One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

$$y=6x+18$$One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?

$$y=6x+18$$

One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?