Problem 15 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-3,\) passing through (-2,-3)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 3 = -3(x + 2)\). The slope-intercept form of the line is \(y = -3x - 6\).

Step 1: Write Equation in Point-Slope form

Use the formula \(y-y_1=m(x-x_1)\), substitute \(m=-3\), \(x_1=-2\), and \(y_1=-3\) into it. The equation becomes \(y - (-3) = -3(x - (-2))\) which simplifies to \(y + 3 = -3(x + 2)\).

Step 2: Write Equation in Slope-Intercept Form

Convert the point-slope form to the slope-intercept form (y=mx+b). Here the slope m is -3. Solve the equation from the previous step for y to identify the y-intercept, b. In this case, it simplifies to \(y = -3x - 6\).

Key Concepts

Understanding Point-Slope Form

The point-slope form of a linear equation provides a straightforward way to write the equation of a line when you're given a point on the line and the slope. This form is expressed as \( y - y_1 = m(x - x_1) \), where:

• \( y_1 \) and \( x_1 \) are the coordinates of a given point on the line.
• \( m \) is the slope of the line.

This format is very useful because it directly incorporates one point on the line and the line's slope. In the given problem, the slope \( m \) is \(-3\) and the point is \((-2, -3)\). By substituting these values into the formula, we get the equation \( y + 3 = -3(x + 2) \).
This formula highlights the linear relationship between \( x \) and \( y \), using the slope and point to define the line's characteristics precisely.

Exploring Slope-Intercept Form

The slope-intercept form is another way to express the equation of a line. This form is represented as \( y = mx + b \), where:

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

This format is particularly helpful for quickly understanding the steepness of the line (through the slope \( m \)) and where it meets the y-axis (the intercept \( b \)). From the point-slope equation \( y + 3 = -3(x + 2) \), we rearrange it to solve for \( y \) by distributing and combining like terms:
\( y = -3x - 6 \).
Here, the slope \( m \) is \(-3\) and the y-intercept \( b \) is \(-6\). This tells us that the line crosses the y-axis at \(-6\), and each step along the x-axis decreases the y value by 3 units.

Connecting with the Equation of a Line

Understanding the equation of a line means grasping how these forms - point-slope and slope-intercept - represent the same line in different ways. Both forms provide valuable insights into the line's behavior:

• The point-slope form is ideal for creating a line equation with a known point and slope.
• The slope-intercept form is excellent for quickly sketching a graph and seeing the starting point and inclination direction.

The ability to convert between these forms enhances your flexibility in solving linear problems and can simplify different tasks, such as graphing or finding intersections. Recognizing this connection also aids in understanding the broader concept of linearity in mathematics.