Find the parallel line of x - y = 4 that passes through the point (4, -2).
Understand the Problem
The question is asking for the equation of a line that is parallel to the given line x - y = 4 and passes through the specified point (4, -2). A parallel line will have the same slope as the original line, so we will first determine the slope from the original line's equation and then use the point to find the new line's equation.
Answer
The equation of the line is $y = x - 6$.
Answer for screen readers
The equation of the line that is parallel to $x - y = 4$ and passes through the point $(4, -2)$ is: $$ y = x - 6 $$
Steps to Solve
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Convert the original line's equation to slope-intercept form
To find the slope of the original line, we first need to convert the given equation $x - y = 4$ into the slope-intercept form, which is $y = mx + b$, where $m$ is the slope.
Starting with the original equation: $$ x - y = 4 $$
Rearranging it: $$ -y = -x + 4 $$
Multiplying through by -1 gives: $$ y = x - 4 $$
Thus, the slope $m$ of the original line is $1$.
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Use the point-slope form to find the new line's equation
Since parallel lines have the same slope, the new line passing through the point $(4, -2)$ will also have the slope $m = 1$. We can use the point-slope form of the equation, which is given by: $$ y - y_1 = m(x - x_1) $$
Here, $(x_1, y_1)$ is the point $(4, -2)$ and $m = 1$. Substituting these values into the equation: $$ y - (-2) = 1(x - 4) $$
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Simplify to get the equation in slope-intercept form
Now we simplify the equation: $$ y + 2 = 1(x - 4) $$
Distributing the $1$: $$ y + 2 = x - 4 $$
Subtracting $2$ from both sides gives us: $$ y = x - 6 $$
This is the equation of the line that is parallel to the original line and passes through the specified point.
The equation of the line that is parallel to $x - y = 4$ and passes through the point $(4, -2)$ is: $$ y = x - 6 $$
More Information
The line $y = x - 6$ has the same slope as the original line, ensuring it is parallel. This method of finding parallel lines by using the same slope is commonly used in coordinate geometry.
Tips
- Not converting the original line's equation correctly to slope-intercept form can lead to finding the wrong slope. To avoid this, carefully rearrange the terms and check that you have isolated $y$ correctly.
- Confusing the point-slope form of the equation can result in mistakes. Always ensure you are substituting the correct values for $x_1$, $y_1$, and $m$.
