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 2 solved the equation correctly.
Student 1 incorrectly solved for a by subtracting x from both sides of the equation, which is not a valid step. This creates a=x-by, not a=yb-x.
Student 3 incorrectly solved for a by subtracting x from both sides of the equation and then multiplying by -1, which is not a valid step. This creates -by+x=a, not a=yb-x.
Student 2 correctly solved the equation by multiplying both sides by -1 to isolate a. This results in -yb+x=a, which is the correct solution.
Student 2 solved the equation correctly.
In the original equation, y = x - ab, all three students tried to isolate "a" on one side of the equation.
Student 1 made a mistake by incorrectly distributing "b" in the second step, obtaining by = x - a instead of by = x - ab. This mistake resulted in by - x = a instead of by - x = ab, therefore solving for "a" incorrectly.
Student 3 made the same mistake as Student 1. They incorrectly distributed "b" in the second step, again resulting in by - x = a instead of by - x = ab. This also led to solving for "a" incorrectly.
On the other hand, Student 2 correctly distributed "b" in the second step, resulting in yb = x - a. Then they correctly isolated "a" by subtracting x from both sides, obtaining yb - x = -a. Finally, they multiplied both sides by -1 to solve for "a", giving -yb + x = a.
Therefore, Student 2 solved the equation correctly by correctly isolating "a" and obtaining the correct expression for "a".
To determine which student solved the equation correctly, let's analyze the work of each student and compare it to the original equation.
Student 1 substituted the expression "x - a" back into the original equation for "by" correctly. However, they made an error when isolating "a" by subtracting "x" from both sides. The correct step would be to add "x" to both sides, giving us "by + x = a". Therefore, Student 1 did not solve the equation correctly.
Student 2 made a similar mistake to Student 1. They substituted the expression "x - a" for "yb" correctly but made an error when isolating "a" by subtracting "x" from both sides. The correct step would be to add "x" to both sides, resulting in "-yb + x = a". Therefore, Student 2 also did not solve the equation correctly.
Student 3, on the other hand, correctly substituted the expression "x - a" for "by". Then, they correctly isolated "a" by subtracting "by" from both sides. The final step of multiplying "-1" to both sides to change the sign is also fine. Thus, Student 3 correctly solved the equation.
Therefore, Student 3 is the only student who solved the equation correctly.