Find the parallel line of y = 3x + 1 that passes through the point (3, -3).

Understand the Problem

The question is asking for the equation of a line that is parallel to the given line y = 3x + 1 and passes through the point (3, -3). To find this, we need to use the same slope as the original line and apply the point-slope form of a line equation.

Answer

The equation is $y = 3x - 12$.

Steps to Solve

Identify the slope of the given line

The slope of the given line $y = 3x + 1$ can be identified from the equation, which is in the slope-intercept form $y = mx + b$. Here, $m$ is the slope. Therefore, the slope is $3$.

Use the point-slope form of a line

To find the equation of the line that is parallel to the original line and passes through the point $(3, -3)$, we use the point-slope form of the equation, which is: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1)$ is the given point and $m$ is the slope. Plugging in the values we have: $$y - (-3) = 3(x - 3)$$

Simplify the equation

Now simplify the equation from the previous step: $$y + 3 = 3(x - 3)$$

Distributing the $3$ gives: $$y + 3 = 3x - 9$$

Isolate y to obtain the slope-intercept form

Subtract $3$ from both sides to isolate $y$: $$y = 3x - 9 - 3$$ This simplifies to: $$y = 3x - 12$$

The equation of the line that is parallel to $y = 3x + 1$ and passes through the point $(3, -3)$ is $y = 3x - 12$.

More Information

This answer shows that two lines that are parallel have the same slope. The new line being determined has a y-intercept different from the original line, which is why it does not intersect but remains parallel.

Tips

Forgetting that parallel lines have the same slope; it's crucial to maintain the slope from the original line.
Misplacing the coordinates when using the point in the point-slope formula. Always double-check that $(x_1, y_1)$ accurately represents the point through which the new line passes.

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