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$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

","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-advanced-math-multiple-choice-question-133"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

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

Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. Therefore, $(x,y)$ is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, $y=3x+a$, from the first equation, $y=2x^{2}-21x+64$. This yields $y-y=\\left(2x^{2}-21x+64\\right)-(3x+a)$, or $0=2x^{2}-24x+64-a$. Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form $rx^{2}+sx+t=0$, where $r$, $s$, and $t$ are constants, has one solution if and only if the discriminant, $s^{2}-4rt$, is equal to zero. Substituting $2$ for $r$, $-24$ for $s$, and $-a+64$ for $t$ in the expression $s^{2}-4rt$ yields $(-24)^{2}-(4)(2)(64-a)$. Setting this expression equal to zero yields $(-24)^{2}-(4)(2)(64-a)=0$, or $8a+64=0$. Subtracting $64$ from both sides of this equation yields $8a=-64$. Dividing both sides of this equation by $8$ yields $a=-8$. Substituting $-8$ for $a$ in the equation $0=2x^{2}-24x+64-a$ yields $0=2x^{2}-24x+64+8$, or $0=2x^{2}-24x+72$. Factoring $2$ from the right-hand side of this equation yields $0=2\\left(x^{2}-12x+36\\right)$. Dividing both sides of this equation by $2$ yields $0=x^{2}-12x+36$, which is equivalent to $0=(x-6)(x-6)$, or $0=\\left(x-6\\right)^{2}$. Taking the square root of both sides of this equation yields $0=x-6$. Adding $6$ to both sides of this equation yields $x=6$.

Choice A is incorrect. This is the value of $a$, not $x$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

"},"encodingFormat":"text/markdown","text":"

$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

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

$-8$

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

—6

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$8$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$6$

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

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-133"}

$$y=2x^{2}-21x+64$$ $$y=3x+a$$In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

$$y=2x^{2}-21x+64$$ $$y=3x+a$$In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

$$y=2x^{2}-21x+64$$ $$y=3x+a$$In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?

$$y=2x^{2}-21x+64$$ $$y=3x+a$$

In the given system of equations, $a$ is a constant. The graphs of the equations in the given system intersect at exactly one point, $(x,y)$, in the $xy$-plane. What is the value of $x$?