Question
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.
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.
Answers
GPT-4o mini
Let's analyze the problem and the viewer's solution.
The equations given are:
1. \( x + y = 20 \) (the total of correct and incorrect answers)
2. \( 2x - y = 28 \) (the scoring equation, which should actually be \( 2x - y = 28\))
The viewer's process included the following steps:
1. They properly recognized that \( x + y = 20 \).
2. They considered the other equation about the points scored, but they incorrectly simplified it as \( -2x - y = -28 \) instead of keeping it as \( 2x - y = 28 \). The correct form would maintain one of the variables as positive while the other is appropriately adjusted, rather than introducing a negative.
The viewer's addition of \( x + y = 20 \) and \( -2x - y = -28 \) should result in:
- \( x + y + (-2x - y) = 20 + (-28) \)
- That simplifies to \( -x = -8 \), thus \( x = 8 \), which is correct for the value of \( x \).
Now let's re-examine the second equation:
If we were to actually apply the scores formula as \( 2x - y = 28 \), we need to properly translate the negative version of it:
- The two equations are therefore:
1. \( x + y = 20 \)
2. \( 2x - y = 28 \)
After solving them simultaneously, one would replace or eliminate as needed leading to a valid solution, but they directly substituted incorrectly.
The viewer didn't actually calculate incorrectly concerning \( x \) and \( y \) arriving at correct answers but instead misapplied the transformation of the second equation while trying to apply negative adjustments.
To conclude, **The viewer mistakenly determined that one of the equations in the system should be \( 2x + y = 28 \). Instead, modifications should have been made before applying the equation.** Thus the correct system should have been utilized all along allowing complete correctness without modification confusion.
The equations given are:
1. \( x + y = 20 \) (the total of correct and incorrect answers)
2. \( 2x - y = 28 \) (the scoring equation, which should actually be \( 2x - y = 28\))
The viewer's process included the following steps:
1. They properly recognized that \( x + y = 20 \).
2. They considered the other equation about the points scored, but they incorrectly simplified it as \( -2x - y = -28 \) instead of keeping it as \( 2x - y = 28 \). The correct form would maintain one of the variables as positive while the other is appropriately adjusted, rather than introducing a negative.
The viewer's addition of \( x + y = 20 \) and \( -2x - y = -28 \) should result in:
- \( x + y + (-2x - y) = 20 + (-28) \)
- That simplifies to \( -x = -8 \), thus \( x = 8 \), which is correct for the value of \( x \).
Now let's re-examine the second equation:
If we were to actually apply the scores formula as \( 2x - y = 28 \), we need to properly translate the negative version of it:
- The two equations are therefore:
1. \( x + y = 20 \)
2. \( 2x - y = 28 \)
After solving them simultaneously, one would replace or eliminate as needed leading to a valid solution, but they directly substituted incorrectly.
The viewer didn't actually calculate incorrectly concerning \( x \) and \( y \) arriving at correct answers but instead misapplied the transformation of the second equation while trying to apply negative adjustments.
To conclude, **The viewer mistakenly determined that one of the equations in the system should be \( 2x + y = 28 \). Instead, modifications should have been made before applying the equation.** Thus the correct system should have been utilized all along allowing complete correctness without modification confusion.