Determine the value of ''s'' for which the equations ` 5 x + 35 = 60 x + s ` has infinite number of solutions

Options

A. 420
B. 440
C. 460
D. 480

Answer

To determine the value of \( s \) for which the equation \( 5x + 35 = 60x + s \) has an infinite number of solutions, we can follow these steps:

### Step 1: Understand the condition for infinite solutions
For the equation \( 5x + 35 = 60x + s \) to have an infinite number of solutions, the coefficients of \( x \) on both sides must be proportional to the constant terms. This means that the ratio of the coefficients of \( x \) must equal the ratio of the constant terms.

### Step 2: Set up the ratios
The coefficients of \( x \) are \( 5 \) and \( 60 \). The constant terms are \( 35 \) and \( s \). Therefore, we can set up the following proportion:
\[ \frac{5}{60} = \frac{35}{s} \]

### Step 3: Simplify the left side
The left side simplifies as follows:
\[ \frac{5}{60} = \frac{1}{12} \]
So we have:
\[ \frac{1}{12} = \frac{35}{s} \]

### Step 4: Cross-multiply to solve for \( s \)
Cross-multiplying gives us:
\[ 1 \cdot s = 12 \cdot 35 \]
This simplifies to:
\[ s = 12 \cdot 35 \]

### Step 5: Calculate \( 12 \cdot 35 \)
Now, we calculate \( 12 \cdot 35 \):
\[ 12 \cdot 35 = 420 \]

### Conclusion
Thus, the value of \( s \) for which the equation has an infinite number of solutions is:
\[ \boxed{420} \]

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