Question

problems 7–8: maneli and trinidad are trying to solve the equation \\(\frac{2}{3} + x = \frac{1}{3}\\). - maneli says: i think we should multiply each side by \\(\frac{3}{2}\\) because that is the reciprocal of \\(\frac{2}{3}\\).

Explanation:

Step1: Identify correct solving method

To solve $\frac{2}{3} + x = \frac{1}{3}$, we isolate $x$ by subtracting $\frac{2}{3}$ from both sides, not multiplying by the reciprocal (this method is for equations where $x$ is multiplied by a fraction, not added to one).

Step2: Isolate the variable $x$

Subtract $\frac{2}{3}$ from both sides:
$x = \frac{1}{3} - \frac{2}{3}$

Step3: Calculate the value of $x$

Compute the subtraction:
$x = \frac{1-2}{3} = -\frac{1}{3}$

Step4: Evaluate Maneli's method

If we multiply each side by $\frac{3}{2}$, we get:
$\frac{3}{2} \times \frac{2}{3} + \frac{3}{2}x = \frac{3}{2} \times \frac{1}{3}$
$1 + \frac{3}{2}x = \frac{1}{2}$
This does not isolate $x$ directly and is an incorrect first step for this equation.

Answer:

Maneli's method is incorrect. The correct solution for the equation is $x = -\frac{1}{3}$, found by subtracting $\frac{2}{3}$ from both sides of the original equation.