Question

find the equation and slope of a line that is perpendicular to the y - axis and passes through the point (-2, 3). click on undefined as the answer if necessary. equation: \boxed{} slope: \boxed{}

Explanation:

Step1: Analyze the line perpendicular to y - axis

A line perpendicular to the \(y\) - axis is a horizontal line? No, wait, a line perpendicular to the \(y\) - axis is a horizontal line? No, actually, the \(y\) - axis is vertical (with an undefined slope). A line perpendicular to a vertical line is a horizontal line? No, correction: The \(y\) - axis is vertical, so a line perpendicular to the \(y\) - axis is horizontal? Wait, no. The slope of the \(y\) - axis is undefined (since it's a vertical line, the formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\) has a denominator of 0). A line perpendicular to a vertical line ( \(y\) - axis) is a horizontal line? No, wait, vertical and horizontal lines are perpendicular. Wait, the \(y\) - axis is vertical (undefined slope), a line perpendicular to it is horizontal, with slope \(m = 0\)? Wait, no, let's recall: The slope of a vertical line (like \(x=a\)) is undefined. The slope of a horizontal line (like \(y = b\)) is 0. And a vertical line and a horizontal line are perpendicular. So a line perpendicular to the \(y\) - axis (which is vertical) is a horizontal line. Wait, but the problem says "perpendicular to the \(y\) - axis". Wait, maybe I made a mistake. Wait, the \(y\) - axis is vertical, so a line perpendicular to the \(y\) - axis is horizontal, so it has a slope of 0? Wait, no, wait: Let's think about the direction. The \(y\) - axis is vertical (along the \(y\) - direction). A line perpendicular to it would be horizontal (along the \(x\) - direction), so it's a horizontal line, which has the equation \(y = k\), where \(k\) is a constant, and its slope is 0. But wait, the problem says "passes through the point \((- 2,3)\)". Wait, no, wait, maybe I mixed up perpendicular to \(x\) - axis and \(y\) - axis. Wait, the \(x\) - axis is horizontal (slope 0), a line perpendicular to \(x\) - axis is vertical (undefined slope). The \(y\) - axis is vertical (undefined slope), a line perpendicular to \(y\) - axis is horizontal (slope 0). Wait, but let's check the point \((-2,3)\). If the line is perpendicular to the \(y\) - axis (so horizontal), then the equation of the line is \(y = 3\), and the slope is 0? Wait, no, wait, maybe I got it wrong. Wait, no, the \(y\) - axis is vertical. A line perpendicular to a vertical line is horizontal. So the line we want is horizontal, passing through \((-2,3)\). So the equation of a horizontal line is \(y=k\), where \(k\) is the \(y\) - coordinate of any point on the line. Since the line passes through \((-2,3)\), the \(y\) - coordinate is 3, so the equation is \(y = 3\), and the slope of a horizontal line is 0. Wait, but wait, the problem says "perpendicular to the \(y\) - axis". Wait, maybe I made a mistake. Let's re - express: The \(y\) - axis has an undefined slope (vertical line). A line perpendicular to a vertical line is a horizontal line (slope 0). So the line is horizontal, passing through \((-2,3)\), so \(y = 3\), slope \(m = 0\). Wait, but let's confirm with the slope formula. For a horizontal line, any two points on the line have the same \(y\) - coordinate. Let's take two points on \(y = 3\), say \((x_1,3)\) and \((x_2,3)\). Then the slope \(m=\frac{3 - 3}{x_2 - x_1}=\frac{0}{x_2 - x_1}=0\) (as long as \(x_2 eq x_1\)). And since it's perpendicular to the \(y\) - axis (vertical), this makes sense.

Wait, but the problem says "perpendicular to the \(y\) - axis". So the line is horizontal, equation \(y = 3\), slope \(m = 0\). Wait, but let's check the point \((-2,3)\). If the line is horizontal (perpendicular to \(y\) - axis), then for all \(x\), \(y = 3\), so when \(x=-2\), \(y = 3\), which matches the point.

Step2: Write the equation and slope

The slope of a horizontal line (perpendicular to \(y\) - axis) is \(m = 0\). The equation of a horizontal line passing through the point \((x_0,y_0)\) is \(y=y_0\). Here, \(y_0 = 3\), so the equation is \(y = 3\), and the slope \(m = 0\).

Answer:

Equation: \(y = 3\)
Slope: \(0\)