Questions: Add. (4z+1)/(3z) + (5z-13)/z Simplify your answer as much as possible.

Add. (4z+1)/(3z) + (5z-13)/z Simplify your answer as much as possible.
Transcript text: Add. \[ \frac{4 z+1}{3 z}+\frac{5 z-13}{z} \] Simplify your answer as much as possible.
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Solution

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Solution Steps

To add the given fractions, we need to find a common denominator. The denominators are \(3z\) and \(z\). The least common denominator (LCD) is \(3z\). Rewrite each fraction with the LCD, then add the numerators. Finally, simplify the resulting expression if possible.

Step 1: Finding a Common Denominator

To add the fractions \(\frac{4z + 1}{3z}\) and \(\frac{5z - 13}{z}\), we first identify the least common denominator (LCD), which is \(3z\).

Step 2: Rewriting the Fractions

We rewrite the second fraction to have the common denominator: \[ \frac{5z - 13}{z} = \frac{(5z - 13) \cdot 3}{z \cdot 3} = \frac{15z - 39}{3z} \]

Step 3: Adding the Fractions

Now we can add the two fractions: \[ \frac{4z + 1}{3z} + \frac{15z - 39}{3z} = \frac{(4z + 1) + (15z - 39)}{3z} = \frac{19z - 38}{3z} \]

Step 4: Simplifying the Expression

The expression \(\frac{19z - 38}{3z}\) can be factored: \[ \frac{19(z - 2)}{3z} \]

Final Answer

Thus, the simplified form of the given expression is: \[ \boxed{\frac{19(z - 2)}{3z}} \]

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