Problem 10 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given point and has the given slope. $$ (2,-3), m=0 $$

Step-by-Step Solution

Answer

The equation of the line that passes through the point (2,-3) and has a slope of 0 is \(y = -3\).

Step 1: Identify the given point and slope

The given point is (2,-3), so \(x_1 = 2\) and \(y_1 = -3\). The given slope \(m = 0\).

Step 2: Substitute into the y-intercept formula

Substituting \(x_1\), \(y_1\), and \(m\) into \(b = y_1 - mx_1\) gives \(b = -3 - 0 \times 2 = -3\).

Step 3: Write the equation in slope-intercept form

Substitute \(m\) and \(b\) into the slope-intercept formula \(y = mx + b\) to get the equation of the line: \(y = 0x - 3\). Normally, we simplify this to \(y = -3\).

Key Concepts

Equation of a Line

When you're tasked with finding the equation of a line in mathematics, especially in its slope-intercept form, your main goal is to identify key components that help form the equation. The slope-intercept formula of a line is expressed as:

• \( y = mx + b \)

In this formula:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

By substituting the slope and y-intercept into the slope-intercept form, you can derive the line's equation. For example, if you're given a point on the line and the slope, you can first find the y-intercept and then use these values to form the complete equation of the line in slope-intercept form.

Slope

The slope of a line, denoted as \( m \), describes the direction and steepness of the line. It is calculated as the "rise over run," which means the vertical change divided by the horizontal change between two points on the line. In mathematical terms, if you have two points on a line (\( x_1, y_1 \)) and (\( x_2, y_2 \)), the slope \( m \) is determined by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope means the line rises as it moves from left to right, while a negative slope indicates the line falls. A zero slope, as in the original exercise, means the line is perfectly horizontal. This implies that no matter the value of \( x \), the \( y \) value remains constant. In our example, since the slope \( m = 0 \), the equation of the line reflects a consistent \( y \)-value regardless of \( x \). This results in a horizontal line expressed by the equation \( y = -3 \).

Y-Intercept

The y-intercept is a crucial element in the equation of a line. It represents the point where the line crosses the y-axis. This is the "\( b \)" in the slope-intercept form equation \( y = mx + b \). To find the y-intercept, you need a point on the line and the slope of the line. Using the formula:

• \( b = y_1 - mx_1 \)

You substitute the coordinates of the given point \((x_1, y_1)\) and the slope \( m \) into the equation. For instance, in the original exercise, by substituting \( x_1 = 2 \), \( y_1 = -3 \), and \( m = 0 \), we find:

• \( b = -3 - 0 \times 2 = -3 \)

Here, \( b = -3 \) means the line crosses the y-axis at \(-3\). In the resulting equation \( y = -3 \), the y-intercept explicitly defines the entire behavior of the line because the slope is zero, indicating the line will always intersect the y-axis at this constant value.