In the xy-plane, the equations x + 2y = 10 and 3x + 6y = c represent the same line for some constant c. What is the value of c ?

Explanation

To find the value of \( c \) such that the equations \( x + 2y = 10 \) and \( 3x + 6y = c \) represent the same line, we can follow these steps:

### Step 1: Understand the relationship between the equations
The two equations represent the same line if one can be obtained from the other by multiplying or dividing by a constant.

### Step 2: Rewrite the second equation
We start with the second equation:
\[ 3x + 6y = c \]
We can factor out a 3 from the left-hand side:
\[ 3(x + 2y) = c \]

### Step 3: Compare with the first equation
Now, we can express the equation as:
\[ x + 2y = \frac{c}{3} \]
We need this equation to be equivalent to the first equation \( x + 2y = 10 \).

### Step 4: Set the right-hand sides equal
Since both equations must be equal, we set their right-hand sides equal to each other:
\[ \frac{c}{3} = 10 \]

### Step 5: Solve for \( c \)
To find \( c \), we multiply both sides of the equation by 3:
\[ c = 10 \times 3 \]
\[ c = 30 \]

### Conclusion
Thus, the value of \( c \) is \( 30 \).

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