Students were asked to solve the equation y = x -a/b for a. Their work is shown below. Which student solved the equation correctly? Justify your answer by explaining why the other two students solved the equation incorrectly.
Student 1
y = x -a / b
by = x -a
by - x = a
Student 2
y = x -a / b
y / b = x -a
y / b - x = -a
- y / b + x = a
Student 3
y = x -a/b
b y = x -a
b y -x = -a
- b y + x = a
Student 1 solved the equation correctly. They correctly isolated the variable a by subtracting x from both sides of the equation.
Student 2 solved the equation incorrectly. They incorrectly distributed the negative sign to both the y and x terms, resulting in a sign error. They also incorrectly isolated the variable a by subtracting x from both sides.
Student 3 solved the equation incorrectly. They incorrectly distributed the negative sign to both the y and x terms, resulting in a sign error. They also incorrectly isolated the variable a by subtracting x from both sides.
Show your work in your answers
Student 1:
Starting equation: y = x - a/b
Multiply both sides by b: by = bx - a
Add x to both sides: by + x = bx - a
Subtract bx from both sides: by - bx + x = -a
Rearrange: x - bx + by = -a
Simplify: (1 - b)x + by = -a
Therefore, a = -[(1 - b)x + by]
Student 2:
Starting equation: y = x - a/b
Multiply both sides by b: by = bx - a
Add -bx to both sides: by - bx = -a
Rearrange: -bx + by = -a
Rewrite both sides: -bx - by = a
Therefore, a = -(bx + by)
Student 3:
Starting equation: y = x - a/b
Multiply both sides by b: by = bx - a
Add x to both sides: by + x = bx - a
Subtract bx from both sides: by - bx + x = -a
Rearrange: x - bx + by = -a
Simplify: (1-b)x + by = -a
Therefore, a = -[(1-b)x + by]
Student 1 solved the equation correctly.
Explanation:
Student 1 correctly applied the subtraction operation to both sides of the equation, isolating the variable 'a' on one side of the equation.
However, student 2 and student 3 made mistakes in their solutions:
- Student 2 incorrectly distributed the denominator 'b' to both 'y' and 'x' instead of just 'x' in the numerator. This led to an incorrect equation where 'a' was multiplied by '-1'.
- Student 3 also made a similar mistake as student 2 by incorrectly distributing 'b' to both 'y' and 'x' in the numerator. Additionally, they also incorrectly assigned the negative sign to 'a'. The negative sign should be attached to the 'by' term, resulting in a positive value for 'a'.
To determine which student solved the equation correctly, let's analyze the work of each student:
Student 1:
y = x - a / b
by = x - a
by - x = a
Student 2:
y = x - a / b
y / b = x - a
y / b - x = -a
- y / b + x = a
Student 3:
y = x - a / b
b y = x - a
b y - x = -a
- b y + x = a
The correct representation of the equation is: y = x - a / b
Out of the three students, Student 1 solved the equation correctly. They correctly isolated the variable a by subtracting x from both sides of the equation.
Now, let's explain why the other two students solved the equation incorrectly:
Student 2 started correctly by rearranging the equation, but made a mistake when isolating the variable a. They subtracted x from both sides but didn't distribute the negative sign to the denominator. As a result, they ended up with -y/b + x = -a instead of a.
Student 3 made a similar mistake as Student 2. They multiplied both sides of the equation by b correctly, but then when isolating the variable a, they distributed the negative sign incorrectly. They ended up with -by + x = -a instead of a.
Therefore, Student 1 is the only one who correctly solved the equation by isolating the variable a on one side of the equation.