Final answer:
The student's solution x = -27/20 is correct. The equation -2/3(x + 3/5) = 1/2 can be solved by distributing the fraction, finding a common denominator, and multiplying by the reciprocal of the coefficient of x.
Step-by-step explanation:
To solve the equation -2/3(x + 3/5) = 1/2, first distribute the fraction -2/3 across the parentheses. This gives us -2/3 * x - 2/3 * 3/5, which simplifies to -2/3x - 6/15. Simplifying the second term we get -2/3x - 2/5.
Now, converting -2/5 to a fraction with a denominator of 3 to match the first term, we obtain -6/15, which is the same as -2/5. Therefore, the equation further simplifies to -2/3x - 2/5. We then add 2/5 to both sides of the equation to isolate the x term:
-2/3x = 1/2 + 2/5
To add the fractions on the right, we need a common denominator, which is 10. Thus, we convert them:
1/2 = 5/10 and 2/5 = 4/10, giving us 5/10 + 4/10 = 9/10. Now, we replace the fractions on the right side with their sum:
-2/3x = 9/10
Finally, to solve for x, we divide both sides by -2/3 or multiply by -3/2, which is the reciprocal:
x = (9/10) * (-3/2)
Multiplying the numerators and denominators, we get:
x = -27/20
So, the solution to the equation is x = -27/20, which is the answer the student obtained.