Students were asked to solve the equation y = (x - a)/b for a. Their work is shown below.
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; by = x - a; by - x = - a; - b * y + 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 in the last step by subtracting "x" from both sides of the equation instead of adding it to both sides. This mistake resulted in a sign error in the final answer.
Student 2 also made a sign error in the last step by subtracting "a" from both sides of the equation with a negative sign, instead of adding it. This mistake also resulted in a sign error in the final answer.
Student 3 correctly subtracted "x" from both sides of the equation in the second step and then multiplied both sides by "-b" to isolate "a". This resulted in the correct expression for "a" on the right side of the equation. Therefore, Student 3 solved the equation correctly.
Student 1 and Student 2 solved the equation incorrectly.
Student 1's error is in the step "by = x - a". They mistakenly distributed the b to both terms in the numerator, which is not correct. The proper way to write it would be "b * y = x - a".
Student 2's error is also in the step "y/b = x - a". They mistakenly subtracted "a" on both sides, which should only be subtracted from one side of the equation. The correct step would be "y/b + a = x".
Student 3 is the student who solved the equation correctly. They correctly isolated the variable a on one side by subtracting "x" from both sides: "by - x = - a". Then they multiplied both sides by -1 and rearranged the terms to get the correct form: "- b * y + x = a". This is the correct solution for the equation.
To determine which student solved the equation correctly, let's analyze each student's steps:
Student 1: y = (x - a)/b; by = x - a; by - x = a
In this step, Student 1 correctly multiplied both sides of the equation by b to isolate (x - a). However, their error occurs when they subtract x from both sides of the equation. Instead, they should have subtracted a from both sides. Therefore, their final expression is incorrect.
Student 2: y = (x - a)/b; y/b = x - a; y/b - x = - a; - y/b + x = a
Student 2 correctly multiplied both sides of the equation by b to isolate (x - a). However, their mistake occurs when trying to isolate a. They subtract x from both sides, which results in - a on the right side of the equation. This is incorrect because -a should be positive a. Hence, Student 2's final expression is incorrect.
Student 3: y = (x - a)/b; by = x - a; by - x = - a; - b * y + x = a
Student 3 made the same mistake as Student 1 by subtracting x instead of a. Therefore, their final expression is incorrect.
Based on the analysis, none of the students solved the equation correctly. Both Student 1 and Student 3 incorrectly subtracted x instead of a, while Student 2 incorrectly isolated a as -a instead of a.
To solve the equation y = (x - a)/b correctly, the correct approach is:
1. Multiply both sides of the equation by b: b * y = x - a
2. Add a to both sides of the equation: b * y + a = x
3. Finally, switch the sides to isolate a: a = x - b * y
None of the students reached this correct final expression, so it can be concluded that none of them solved the equation correctly.