Need help to find the equation of a line parallel to x - y = 4.
Understand the Problem
The question is asking for the equation of a line that is parallel to the given line x - y = 4. Since parallel lines have the same slope, we need to determine the slope of the given line and then use it to write the equation of a new line that is parallel to it.
Answer
The equation of a line parallel to $x - y = 4$ is $y = x - x_0 + y_0$, where $(x_0, y_0)$ is the point through which it passes.
Answer for screen readers
The equation of a line parallel to $x - y = 4$ can be expressed in point-slope form as $y = x - x_0 + y_0$, where $(x_0, y_0)$ is any point through which the line passes.
Steps to Solve
- Rewrite the given line in slope-intercept form
We start by rewriting the equation of the given line to find its slope. The equation is $x - y = 4$. Let's solve for $y$:
$$ -y = -x + 4 $$
Now, multiply both sides by -1:
$$ y = x - 4 $$
From this, we can see that the slope (m) of the given line is 1.
- Use the slope to write the parallel line equation
Parallel lines share the same slope. Therefore, the slope of our new line is also 1. Let's assume the new line should pass through a point $(x_0, y_0)$. The equation in point-slope form is:
$$ y - y_0 = m(x - x_0) $$
Substituting the slope and the specific point gives us:
$$ y - y_0 = 1(x - x_0) $$
- Simplify the equation
Now we can simplify this equation further. Distributing gives:
$$ y - y_0 = x - x_0 $$
Finally, rearranging gives us:
$$ y = x - x_0 + y_0 $$
This represents the equation of the line parallel to the given line that passes through the point $(x_0, y_0)$.
The equation of a line parallel to $x - y = 4$ can be expressed in point-slope form as $y = x - x_0 + y_0$, where $(x_0, y_0)$ is any point through which the line passes.
More Information
The slope of the line $y = x - 4$ is 1, which indicates that any line parallel to it will also have a slope of 1. The intercept can vary depending on the point the new line passes through.
Tips
- Confusing the slope with the y-intercept. Remember, the slope indicates the steepness of the line, whereas the y-intercept is where the line crosses the y-axis.
- Forgetting that parallel lines must have the same slope, which is crucial in writing the new line's equation accurately.
