Find the Line Equation: Parallel to x+3y=4x+9 Through Point (6,14)

To solve the problem of determining the equation of the line parallel to $x + 3y = 4x + 9$ and passing through point $(6, 14)$, we will follow these steps:

• Step 1: Rearrange the given line to find its slope.
• Step 2: Use the slope and the given point to apply the point-slope form of a line.
• Step 3: Convert the equation into the slope-intercept form for clarity.

Let's perform each step:

Step 1: First, the equation $x + 3y = 4x + 9$ simplifies to find the slope. Subtract $x$ from both sides to get $3y = 3x + 9$. By dividing everything by 3, this gives $y = x + 3$. Therefore, the slope, $m$, is 1.

Step 2: Given that parallel lines share the same slope, the slope of the desired line is also 1. Applying the point-slope form: $y - 14 = 1(x - 6)$.

Step 3: Simplifying this equation yields $y - 14 = x - 6$. Further rearranging gives $y = x + 8$.

Thus, the equation of the line parallel to the given line and passing through $(6, 14)$ is $y = x + 8$.