Question

In Exercises \(32-37\), solve each equation for the unknown quantity. Check your answers. $$\frac{2}{3}+x=-\frac{1}{7}$$

Short answer

#tag_title# Question Solve the equation and find the value of x: $$\frac{2}{3} + x = -\frac{1}{7}$$

Step-by-step solution

Identify the equation to be solved

The given equation is $$\frac{2}{3} + x = -\frac{1}{7}.$$

Isolate x

To isolate x, we need to subtract \(\frac{2}{3}\) from both sides of the equation: $$x = -\frac{1}{7} - \frac{2}{3}.$$

Find a common denominator and simplify

The common denominator of 3 and 7 is 21. We need to rewrite both fractions with the common denominator: $$x = -\frac{3}{21} - \frac{14}{21}.$$ Now, we can subtract the numerator values and simplify the fraction: $$x = -\frac{3-14}{21} = -\frac{17}{21}.$$

Check the solution

To check the solution, we will substitute \(x = -\frac{17}{21}\) back into the original equation to see if it holds true: $$\frac{2}{3} + x = -\frac{1}{7}$$ $$\frac{2}{3} - \frac{17}{21} = -\frac{1}{7}$$ $$-\frac{1}{21}=\frac{-1}{7}$$ $$-\frac{1}{21}\neq\frac{-1}{7}$$ We made a mistake when we subtracted \(\frac{14}{21}\) from \(\frac{3}{21}\), it should be subtracted from \(\frac{-3}{21}\).

Correct the error and simplify

We will now correct the error and complete the calculation: $$x = -\frac{-3}{21} - \frac{14}{21}$$ $$x = -\frac{-3 - 14}{21} = -\frac{-17}{21}.$$