Students were asked to solve the equation y=x−ab for a. Their work is shown below.
Student 1:
y=x−ab
by=x−a
by−x=a
Student 2:
y=x−ab
yb=x−a
yb−x=−a
−yb+x=a
Student 3:
y=x−ab
by=x−a
by−x=−a
−by+x=a
Which student solved the equation correctly? Justify your answer by explaining why the other two students solved the equation incorrectly.
Student 3 solved the equation correctly.
Student 1 made a mistake by incorrectly subtracting "a" from both sides of the equation, resulting in the equation "by - x = a" instead of "by - x = -a".
Student 2 also made a mistake by incorrectly dividing both sides of the equation by "b", resulting in the equation "-yb + x = a" instead of "-yb + x = -a".
Student 3, on the other hand, correctly subtracted "a" from both sides of the equation and obtained the correct equation "-by + x = a".
Student 3 solved the equation correctly.
In Student 1's work, they correctly isolated the "a" term on one side of the equation, but they made a mistake when they switched the sign of "a" and "x" on the other side of the equation. This error led to an incorrect result.
In Student 2's work, they made a similar mistake to Student 1. They correctly isolated the "a" term on one side of the equation, but they switched the sign of both "a" and "x" on the other side of the equation. This error also led to an incorrect result.
On the other hand, Student 3 correctly isolated the "a" term on one side of the equation and kept the signs of "a" and "x" unchanged on the other side of the equation. This approach ensures that the equation is solved correctly and gives the correct result.
To determine which student solved the equation correctly, let's analyze each student's work:
Student 1:
y = x - ab
by = x - a
by - x = a
Student 2:
y = x - ab
yb = x - a
yb - x = -a
-yb + x = a
Student 3:
y = x - ab
by = x - a
by - x = -a
-by + x = a
To solve the equation correctly, we need to isolate the variable "a". We can see that Student 2 and Student 3 made a mistake by applying the distributive property incorrectly.
Student 2 wrongly distributed the negative sign when finding the value of "a". Instead of distributing the negative sign, they should have subtracted "x" from both sides to isolate "a". Incorrectly distributing the negative sign led to the incorrect sign in their final equation, -yb + x = a.
Similarly, Student 3 also distributed the negative sign incorrectly when finding the value of "a". Instead of distributing the negative sign, they should have subtracted "x" from both sides to isolate "a". Incorrectly distributing the negative sign led to the incorrect sign in their final equation, -by + x = a.
On the other hand, Student 1 correctly isolated "a" by subtracting both sides by "x". So, Student 1 solved the equation correctly.
In summary, Student 1 solved the equation correctly by properly isolating "a" through subtraction on both sides. The other two students made errors by incorrectly applying the distributive property.