Find the parallel line of x + 2y = 4 that passes through the point (0, -4)
Understand the Problem
The question is asking for the equation of a line that is parallel to the original line given by x + 2y = 4 and that also passes through the specific point (0, -4). To solve this, we first identify the slope of the original line and then use that slope to find the equation of the new line using the point-slope form of a linear equation.
Answer
The equation of the line parallel to the original line is \( y = -\frac{1}{2}x - 4 \).
Answer for screen readers
The equation of the line parallel to the original line is ( y = -\frac{1}{2}x - 4 ).
Steps to Solve
- Convert the original line equation to slope-intercept form
We start with the original line equation: $$ x + 2y = 4 $$
To convert it to slope-intercept form ($y = mx + b$), we rearrange the equation: $$ 2y = -x + 4 $$ Now divide by 2: $$ y = -\frac{1}{2}x + 2 $$
From this, we see that the slope ($m$) of the original line is $-\frac{1}{2}$.
- Use the point-slope form for the new line
Since we want a new line that is parallel to the original line, it will have the same slope of $-\frac{1}{2}$. We will use the point-slope form of a line: $$ y - y_1 = m(x - x_1) $$
Here, $(x_1, y_1) = (0, -4)$ and $m = -\frac{1}{2}$.
Plugging in the values: $$ y - (-4) = -\frac{1}{2}(x - 0) $$
- Simplify the equation
Now we simplify: $$ y + 4 = -\frac{1}{2}x $$ Subtract 4 from both sides: $$ y = -\frac{1}{2}x - 4 $$
This is the equation of the line parallel to the original line that passes through the point (0, -4).
The equation of the line parallel to the original line is ( y = -\frac{1}{2}x - 4 ).
More Information
The line has the same slope as the original line, which is important for parallel lines. The point-slope form is a great tool for finding the equation of a line when you know a point it passes through and its slope.
Tips
- Confusing parallel lines with perpendicular lines. Remember, parallel lines have the same slope.
- Incorrectly substituting the coordinates of the given point in the point-slope form. Always double-check the point values.
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