Contains (5, 1) and is perpendicular to y = 3. write in y=mx+b form

To find the equation of the line that contains the point \((5, 1)\) and is perpendicular to \(y = 3\), we need to follow these steps:

1. **Determine the slope of the given line**: The given equation is \(y = 3\). This is a horizontal line, and its slope \(m\) is 0.

2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of the given line is 0, the slope of the perpendicular line is undefined (it will be a vertical line).

However, since we are required to write the equation in the \(y = mx + b\) form and the slope of the perpendicular line is undefined, we recognize that the perpendicular line must be vertical. The equation of a vertical line through a point \((x_1, y_1)\) is \(x = x_1\).

3. **Write the equation**: The vertical line passes through the point \((5, 1)\), so its equation is:
\[ x = 5 \]

Vertical lines cannot be expressed in the \(y = mx + b\) form since their slope is undefined. However, this is the correct form for a vertical line.

Would you like more details or have any questions about this process?

Here are some related questions for further practice:
1. Write the equation of a line perpendicular to \(y = 2x + 3\) and passing through the point \((4, -1)\).
2. Find the equation of a line parallel to \(y = -5x + 6\) and passing through \((2, 3)\).
3. Determine the equation of the line that passes through the points \((1, 2)\) and \((3, 4)\).
4. Write the equation of a line with slope \(4\) and passing through the point \((0, -2)\).
5. What is the equation of a line with slope \(-\frac{1}{3}\) passing through the point \((6, 5)\)?
6. Find the slope of the line perpendicular to \(y = -7x + 1\).
7. Determine the equation of the line with an undefined slope passing through \((2, -3)\).
8. Write the equation of the horizontal line passing through the point \((4, 2)\).

**Tip:** When finding the equation of a line, always identify the slope first. This will guide you in determining whether the line is horizontal, vertical, or has a specific angle.

Learn how to find the equation of a line that is perpendicular to y = 3 and passes through the point (5, 1). This problem involves understanding slopes, perpendicular lines, and basic algebraic principles. Suitable for students in Grades 7-9.

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