A line has point (0,-3) and is perpendicular to a line with slope 2. What is equation

To find the equation of a line that is perpendicular to a line with slope \(2\) and passes through the point \((0, -3)\), follow these steps:

### Step 1: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of the given line is \(2\), the slope \(m\) of the perpendicular line is:
\[
m = -\frac{1}{2}
\]

### Step 2: Use the point-slope form of the equation of a line
The point-slope form of a line's equation is:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Given that the slope is \(-\frac{1}{2}\) and the point is \((0, -3)\), substitute these values into the equation:
\[
y - (-3) = -\frac{1}{2}(x - 0)
\]
This simplifies to:
\[
y + 3 = -\frac{1}{2}x
\]

### Step 3: Solve for \(y\)
To write the equation in slope-intercept form (\(y = mx + b\)):
\[
y = -\frac{1}{2}x - 3
\]

Thus, the equation of the line is:
\[
y = -\frac{1}{2}x - 3
\]

### Conclusion:
The equation of the line that passes through \((0, -3)\) and is perpendicular to a line with slope \(2\) is:
\[
y = -\frac{1}{2}x - 3
\]

Would you like more details or have any questions?

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Here are 5 related questions for practice:
1. What is the equation of a line that passes through the point \((1, 2)\) and is parallel to the line \(y = 3x - 1\)?
2. Find the equation of a line perpendicular to \(y = -\frac{1}{3}x + 4\) and passing through \((2, 5)\).
3. If two lines are perpendicular, what is the relationship between their slopes?
4. Determine the slope of a line parallel to \(y = 5x + 7\).
5. How do you find the x- and y-intercepts of a line given its equation?

**Tip:** When two lines are perpendicular, their slopes multiply to \(-1\).

A line has point (0,-3) and is perpendicular to a line with slope 2. What is the equation?

Discover how to find the equation of a line that passes through the point (0, -3) and is perpendicular to a line with slope 2. This problem covers key algebra concepts such as slope relationships and the point-slope form of linear equations. Perfect for students in Grades 8-10.

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