If x and y are directly poportional, find the values of `x_1,x_2 and y_1` in the table given below.

Answer

To solve the problem where \( x \) and \( y \) are directly proportional, we will follow these steps:

### Step 1: Understand Direct Proportionality
Since \( x \) and \( y \) are directly proportional, we can express this relationship as:
\[ \frac{x}{y} = k \]
where \( k \) is a constant.

### Step 2: Find the Constant \( k \)
From the given values in the table, we can find \( k \) using the first pair of \( x \) and \( y \):
- Given \( x = 3 \) and \( y = 36 \):
\[ k = \frac{x}{y} = \frac{3}{36} = \frac{1}{12} \]

### Step 3: Calculate \( x_1 \)
Now, we need to find \( x_1 \) when \( y_1 = 60 \):
Using the relationship:
\[ \frac{x_1}{60} = k \]
Substituting the value of \( k \):
\[ \frac{x_1}{60} = \frac{1}{12} \]
Cross-multiplying gives:
\[ x_1 = 60 \times \frac{1}{12} = 5 \]

### Step 4: Calculate \( x_2 \)
Next, we find \( x_2 \) when \( y_2 = 96 \):
Using the relationship:
\[ \frac{x_2}{96} = k \]
Substituting the value of \( k \):
\[ \frac{x_2}{96} = \frac{1}{12} \]
Cross-multiplying gives:
\[ x_2 = 96 \times \frac{1}{12} = 8 \]

### Step 5: Calculate \( y_1 \)
Finally, we need to find \( y_1 \) when \( x = 10 \):
Using the relationship:
\[ \frac{10}{y_1} = k \]
Substituting the value of \( k \):
\[ \frac{10}{y_1} = \frac{1}{12} \]
Cross-multiplying gives:
\[ 10 \times 12 = y_1 \implies y_1 = 120 \]

### Summary of Values
- \( x_1 = 5 \)
- \( x_2 = 8 \)
- \( y_1 = 120 \)

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