ger, simplified fraction, or decimal rounded to two decimal places. $$ \frac{7}{2} z+\frac{13}{7} z-\frac{15}{14}=\frac{5}{7} $$

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UpStudy AI · Accepted Answer

**Step 1: Combine like terms**

The equation is  
$$
\frac{7}{2} z + \frac{13}{7} z - \frac{15}{14} = \frac{5}{7}
$$  
Combine the $$ z $$ terms by finding the sum of their coefficients. To do this, express each coefficient with a common denominator (which is $$14$$):

$$
\frac{7}{2} = \frac{7 \times 7}{2 \times 7} = \frac{49}{14}, \quad \frac{13}{7} = \frac{13 \times 2}{7 \times 2} = \frac{26}{14}
$$

So, the combined coefficient is:
$$
\frac{49}{14} + \frac{26}{14} = \frac{75}{14}
$$

The equation becomes:
$$
\frac{75}{14} z - \frac{15}{14} = \frac{5}{7}
$$

**Step 2: Eliminate the denominators**

Multiply every term by $$14$$ to clear the fractions:
$$
14 \left(\frac{75}{14} z\right) - 14 \left(\frac{15}{14}\right) = 14 \left(\frac{5}{7}\right)
$$
This simplifies to:
$$
75z - 15 = 10
$$

**Step 3: Solve for $$ z $$**

Add $$15$$ to both sides:
$$
75z = 10 + 15 = 25
$$

Divide both sides by $$75$$:
$$
z = \frac{25}{75} = \frac{1}{3}
$$

**Step 4: Express the result**

The solution to the equation is  
$$
z = \frac{1}{3}
$$

As a decimal rounded to two decimal places,  
$$
z \approx 0.33
$$
$$ z = \frac{1}{3} $$ or approximately $$ 0.33 $$.

ger, simplified fraction, or decimal rounded to two decimal places.

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