Question

FOR 14 POINTS Thalia, a quiz show contestant, answered 20 questions. She was awarded 2 points for every correct answer that she gave, but she was penalized 1 point for every incorrect answer. In the end, Thalia had a total of 28 points. In order to determine the number of correct and incorrect answers that she gave, a viewer at home used the system of linear equations x + y = 20 and 2 x + y = 28, with x being the number of correct answers and y being the number of incorrect answers. The viewer solved the system by using the linear combination method as shown. x + y = 20. + StartFraction Negative 2 x minus y = negative 28 over Negative x = negative 8 EndFraction. StartFraction negative x Over negative 1 EndFraction = StartFraction negative 8 Over negative 1 EndFraction. X = 8. 8 + y = 20. 8 + y minus 8 = 20 minus 8. y = 12. The viewer calculated that the contestant answered 8 questions correctly and 12 questions incorrectly. What went wrong? The viewer mistakenly determined that one of the equations in the system should be x + y = 20.Instead modifications should have been made before applying the equation. The viewer mistakenly determined that one of the equations in the system should be 2 x + y = 28. Instead modifications should have been made before applying the equation. The viewer multiplied the equation 2 x + y = 28 by –1 incorrectly. The viewer added the equations x + y = 20 and Negative 2 x minus y = negative 28 incorrectly.

Answer 1

The viewer mistakenly determined that one of the equations in the system should be 2 x + y = 28. Instead modifications should have been made before applying the equation.

Explanation:

Given

Total Questions = 20

Total Points = 28

Correct Answers = +2 points

Incorrect Answers = -1 point

Required

Spot the error in the viewer's calculation

The viewer represents the correct answers with x and the incorrect ones with y.

This implies that

`x + y = 20`

Also, for every correct answers, she gets 2 points;

This can be represented by 2x

and for every incorrect answers, she gets -1 point

This can be represented by -y

This implies that

`2x - y = 28`

This is where the viewer made a mistake

The viewer's mistake is that; instead of subtracting total incorrect points from the correct points, the viewer added them together;

Solving further

`2x - y = 28`

`x + y = 20`

Add both equations together

`2x + x + y - y = 28 + 20`

`3x = 48`

Divide both sides by 3

`(3x)/(3) = (48)/(3)`

`x = (48)/(3)`

`x = 16`

Substitute
`x = 16` in
`x + y = 20`

`16 + y = 20`

Subtract 16 from both sides

`16 - 16 + y = 20 - 16`

`y = 20 - 16`

`y = 4`

This implies that the contestant answered 16 questions correctly and 4 questions incorrectly