system of equations word problem

Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions, but marked their test papers using different grading systems. Mr. Day awarded 5 points for each correct answer, made no deduction for an incorrect or omitted answer, and then added 10 points to the total. Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each incorrect or omitted answer. Two students in different classes answered the same number of questions correctly and received the same test grade. Determine, using algabraic method, the test grade both students received. Confirm your answer graphically or by constructing a table using your graphing calculator.

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Originally Posted by cityismine

Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions, but marked their test papers using different grading systems. Mr. Day awarded 5 points for each correct answer, made no deduction for an incorrect or omitted answer, and then added 10 points to the total. Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each incorrect or omitted answer. Two students in different classes answered the same number of questions correctly and received the same test grade. Determine, using algabraic method, the test grade both students received. Confirm your answer graphically or by constructing a table using your graphing calculator.

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We have three unknowns so we will need three equations to solve this system.

let C be the number correct; I be the number incorrect; and S the score

Mr Day: $\displaystyle 5c+10=s$
Ms. Knight: $\displaystyle 6c-1.5I=s$
question on quiz: $\displaystyle c+I=20 \iff I=20-c$


setting the first to equal we get

$\displaystyle 5c+10=6c-1.5I$

Now we can sub in I to get

$\displaystyle 5c+10=6c-1.5(20-c) \iff 5c+10=6c-30+1.5c \iff 40=2.5c \iff 16=c$

We can now back sub in to find the other values.

$\displaystyle I=20-16=4$ and $\displaystyle s=5c+10=5(16)+10=90$

Yeah :)

Hey, I was trying to solve this using two equations, but I guess you need three. Thanks for explaining it so thoroughly.

Hello, cityismine!

Quote:

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Mr. Day and Ms. Knight gave the same test consisting of 20 short-answer questions,
. . but marked their test papers using different grading systems.
Mr. Day gave 5 points for each correct answer, made no deduction for an incorrect
. . or omitted answer, and then added 10 points to the total.
Ms. Knight gave 6 points for each correct answer and subtracted 1.5 points for each
. . incorrect or omitted answer.
Two students in different classes answered the same number of questions correctly
. . and received the same test grade.

Determine, using algabraic method, the test grade both students received.
Confirm your answer graphically or by constructing a table using your graphing calculator.

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The student in Mr. Day's class got $\displaystyle x$ answers right.
He got 5 points per correct answer plus 10 points.
His grade was: .$\displaystyle 5x + 10$

The student in Ms. Knight's class got $\displaystyle x$ right and $\displaystyle 20-x$ wrong.
She got 6 points per correct answer: .$\displaystyle 6x$
and lost 1.5 point for each wrong answer: .$\displaystyle 1.5(20-x)$
Her grade was: .$\displaystyle 6x - 1.5(20-x) \:=\:7.5x - 30$

The two grades were equal: .$\displaystyle 5x + 10 \:=\:7.5x - 30 \quad\Rightarrow\quad 2.5x \:=\:40 \quad\Rightarrow\quad x \:=\:16$

Their grade was: .$\displaystyle 5(16) + 10 \:=\:7.5(16) - 30 \:=\:\boxed{90}$

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Graph the lines: .$\displaystyle \begin{array}{ccc}y &=& 5x+10 \\ y&=&7.5x - 30\end{array}$

They intersect at: $\displaystyle (16,\,90)$