For each line described, write an equation in
(a)slope-intercept form, if possible, and
(b)standard form.
through $(2,-10)$, perpendicular to a line with undefined slope.

Verified step by step guidance

1. Identify the slope of the given line. Since the line has an undefined slope, it is a vertical line, which means its equation is of the form \(x = k\) for some constant \(k\).
2. Determine the slope of the line perpendicular to the given vertical line. The perpendicular line to a vertical line is a horizontal line, which has a slope of 0.
3. Use the point-slope form of a line to write the equation of the line passing through the point \((2, -10)\) with slope 0. The point-slope form is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point.
4. Simplify the equation from the point-slope form to slope-intercept form, which is \(y = mx + b\). Since the slope is 0, the equation will simplify to \(y = b\), where \(b\) is the y-coordinate of the point.
5. Rewrite the equation in standard form, which is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A \geq 0\). For a horizontal line, this will typically be \(y = C\).

Key Concepts

Slope of a Line

The slope measures the steepness of a line and is calculated as the ratio of vertical change to horizontal change between two points. Lines with undefined slope are vertical lines, meaning their slope cannot be expressed as a finite number.

Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of each other. However, a line perpendicular to a vertical line (undefined slope) is a horizontal line, which has a slope of zero.

Forms of Linear Equations

Slope-intercept form is y = mx + b, where m is slope and b is y-intercept. Standard form is Ax + By = C, where A, B, and C are integers. Some lines, like vertical lines, cannot be expressed in slope-intercept form but can be in standard form.