Solve for `y : (y)/(5 ) + (y)/(7) = 12 `

Options

A. 70
B. 140
C. 35
D. 105

Answer

To solve the equation \(\frac{y}{5} + \frac{y}{7} = 12\), we will follow these steps:

### Step 1: Write down the equation
The equation is given as:
\[ \frac{y}{5} + \frac{y}{7} = 12 \]

### Step 2: Find the least common multiple (LCM)
To combine the fractions on the left side, we need to find the LCM of the denominators 5 and 7. The LCM of 5 and 7 is:
\[ LCM(5, 7) = 5 \times 7 = 35 \]

### Step 3: Rewrite the fractions with the LCM
Now we will rewrite each fraction with the common denominator of 35:
\[ \frac{y}{5} = \frac{7y}{35} \quad \text{(multiply numerator and denominator by 7)} \]
\[ \frac{y}{7} = \frac{5y}{35} \quad \text{(multiply numerator and denominator by 5)} \]

### Step 4: Combine the fractions
Now we can add the fractions:
\[ \frac{7y}{35} + \frac{5y}{35} = \frac{7y + 5y}{35} = \frac{12y}{35} \]

### Step 5: Set the equation equal to 12
Now we have:
\[ \frac{12y}{35} = 12 \]

### Step 6: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 35:
\[ 12y = 12 \times 35 \]

### Step 7: Calculate the right side
Now calculate \(12 \times 35\):
\[ 12 \times 35 = 420 \]
So we have:
\[ 12y = 420 \]

### Step 8: Solve for \(y\)
Now, divide both sides by 12 to solve for \(y\):
\[ y = \frac{420}{12} \]
Calculating this gives:
\[ y = 35 \]

### Final Answer
Thus, the solution for \(y\) is:
\[ y = 35 \]

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