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