To find the equation of a line that passes through the point $(2,3)$ and is parallel to the line $y = 3x + 4$, follow these steps:

Step 1: Identify the slope of the given line

The equation $y = 3x + 4$ is in slope-intercept form $y = mx + b$, where $m$ is the slope.
Thus, the slope of the given line is: $m = 3$

Step 2: Use the slope for the new line

Since parallel lines have the same slope, the slope of the desired line is also: $m = 3$

Step 3: Apply the point-slope form of a line

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Substituting $(x_1, y_1) = (2, 3)$ and $m = 3$:

$y - 3 = 3(x - 2)$

Step 4: Simplify the equation

Distribute the slope: $y - 3 = 3x - 6$ Add 3 to both sides: $y = 3x - 3$

Final Answer

The equation of the line parallel to $y = 3x + 4$ and passing through the point $(2,3)$ is: $y = 3x - 3$

Would you like to see more details or have any questions?

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Here are 5 related questions:

1. What is the general formula for finding the slope between two points?
2. How do you determine if two lines are perpendicular?
3. Can you write the equation of a line perpendicular to $y = 3x + 4$ passing through $(2,3)$?
4. How do you convert a linear equation into point-slope form from slope-intercept form?
5. What happens if two lines have the same slope and intercept?

Tip: Parallel lines always have equal slopes, but different intercepts (unless they are the same line).