Find the parallel line of y = x - 4 that passes through the point (2, 3).
Understand the Problem
The question is asking us to find a line that is parallel to the given line y = x - 4 and also passes through the point (2, 3). A parallel line will have the same slope as the original line. We can determine the slope from the equation and then use the point-slope form to find the equation of the new line.
Answer
The equation of the parallel line is $y = x + 1$.
Answer for screen readers
The equation of the line that is parallel to $y = x - 4$ and passes through the point $(2, 3)$ is $y = x + 1$.
Steps to Solve
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Identify the slope of the given line The equation of the given line is $y = x - 4$. This is in the slope-intercept form $y = mx + b$, where $m$ represents the slope. Thus, the slope of the line is $m = 1$.
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Use the point-slope form The point-slope form of a line is given by the formula $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We will use the point $(2, 3)$ and the slope $m = 1$.
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Substituting the known values We substitute $x_1 = 2$, $y_1 = 3$, and $m = 1$ into the point-slope formula: $$y - 3 = 1(x - 2)$$
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Simplifying the equation Now, we simplify the equation to find the equation of the new line: $$y - 3 = x - 2$$
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Rearranging to slope-intercept form Finally, we can rearrange the equation to get it into the slope-intercept form $y = mx + b$: $$y = x + 1$$
The equation of the line that is parallel to $y = x - 4$ and passes through the point $(2, 3)$ is $y = x + 1$.
More Information
This result shows that parallel lines have the same slope. The new line being parallel means it won't intersect with the original line. The new line is 5 units above the original line due to the difference in their y-intercepts.
Tips
- Forgetting that parallel lines have the same slope: It's important to check that the slopes are equal.
- Incorrectly applying the point-slope formula: Make sure to substitute the point values correctly.
