Question

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

Short answer

Answer: \(b = \frac{1}{6}\)

Step-by-step solution

Subtract \(\frac{1}{2}\) from both sides

To isolate the variable \(b\) and solve for it, subtract \(\frac{1}{2}\) from both sides so that the constant value cancels out on the left-hand side of the equation. This results in the equation: $$b = \frac{2}{3} - \frac{1}{2}$$

Find a common denominator for the fractions

In order to subtract the fractions on the right-hand side of the equation, we need to find a common denominator. The least common multiple (LCM) of the denominators \(2\) and \(3\) is \(6\). Convert the fractions to have the denominator \(6\). This results in the equation: $$b = \frac{4}{6} - \frac{3}{6}$$

Subtract fractions

Now that we have the same denominator, we can subtract the numerators of the fractions: $$b = \frac{4-3}{6}$$ This simplifies to: $$b = \frac{1}{6}$$

Check the solution

To check whether the solution \(b = \frac{1}{6}\) is correct, plug it back into the original equation, and see if it holds true: $$\frac{1}{2} + \frac{1}{6} = \frac{2}{3}$$ To verify this, we need to convert the fractions on the left-hand side of the equation to a common denominator, in this case \(6\): $$\frac{3}{6} + \frac{1}{6} = \frac{2}{3}$$ Adding the two fractions on the left-hand side and simplifying gives: $$\frac{4}{6} = \frac{2}{3}$$ Since both fractions on either side of the equation can be simplified to \(\frac{2}{3}\), the solution is correct, and $$b = \frac{1}{6}$$