Find the parallel line of x + 2y = 4 that passes through the point (0, -4)

Understand the Problem

The question is asking to find the equation of a line that is parallel to the given line (x + 2y = 4) and that passes through the specific point (0, -4). To solve this, we need to determine the slope of the given line and use it to write the equation of the new line using the point-slope form.

Answer

$$ y = -\frac{1}{2}x - 4 $$

Steps to Solve

Find the slope of the given line

First, we need to express the equation of the given line in slope-intercept form (y = mx + b).

Starting with the equation: $$ x + 2y = 4 $$

We can isolate (y): $$ 2y = -x + 4 $$

Now, divide everything by 2: $$ y = -\frac{1}{2}x + 2 $$

The slope (m) of the given line is (-\frac{1}{2}).

Use the slope for the new line

Since the new line is parallel to the original line, it will have the same slope. Therefore, the slope of the new line is also (m = -\frac{1}{2}).

Apply the point-slope form

Now, we will use the point-slope form of the line equation, which is given by: $$ y - y_1 = m(x - x_1) $$

Where ((x_1, y_1)) is the point ((0, -4)). Substituting in the values: $$ y - (-4) = -\frac{1}{2}(x - 0) $$

Simplify the equation

Now, simplify the equation: $$ y + 4 = -\frac{1}{2}x $$

Subtracting 4 from both sides gives us: $$ y = -\frac{1}{2}x - 4 $$

This is the equation of the line that is parallel to the given line and passes through the point (0, -4).

The equation of the line that is parallel to (x + 2y = 4) and passes through the point ((0, -4)) is $$ y = -\frac{1}{2}x - 4 $$

More Information

The slope of the line indicates it descends as you move from left to right. The intercept at (-4) means the line crosses the y-axis below the origin.

Tips

Mixing up the signs when identifying the slope; remember that parallel lines share the same slope.
Forgetting to substitute the point coordinates correctly into the point-slope form.

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