Problem 67 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. $$ \text { Slope }=-6, \text { passes through }(1,3) $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -6x + 9 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Slope

We know the slope \( m \) of the line is \(-6\). Substituting this into the slope-intercept form gives us \( y = -6x + b \).

Step 3: Use the Given Point to Find the Y-Intercept

The line passes through the point \((1, 3)\). Substitute \( x = 1 \) and \( y = 3 \) into the equation to find \( b \):\[3 = -6(1) + b\]\[3 = -6 + b\]

Step 4: Solve for the Y-Intercept

Rearrange the equation to solve for \( b \):\[3 + 6 = b\]\[b = 9\]

Step 5: Write the Final Equation

Substitute \( b = 9 \) back into the equation from Step 2, giving us the final equation of the line:\[y = -6x + 9\]

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and pivotal for describing straight lines. They are usually written in the form:

• Standard Form: \( Ax + By = C \)
• Slope-Intercept Form: \( y = mx + b \)

Linear equations express the relationship between two variables, typically \( x \) and \( y \), where the graph of the equation will always be a straight line. These equations are very useful in practical problems where you wish to understand how one variable affects another in a linear way. The main characteristic of a linear equation is its constant rate of change, represented by the slope. In this exercise, we explored how to express a line using the slope-intercept form, starting from given conditions like a specific slope and a point through which the line passes. This involves understanding the relationships between components of the equation to find missing values, such as the y-intercept.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is represented by the letter \( m \) in the slope-intercept form equation \( y = mx + b \). The slope can tell you how much \( y \) changes for a given change in \( x \).

• A positive slope indicates an upward slant from left to right.
• A negative slope, like \(-6\), means the line slopes downward from left to right.
• A zero slope implies a horizontal line, showing no change in \( y \) as \( x \) changes.

Slope is calculated as the ratio of the rise (the change in \( y \)) over the run (the change in \( x \)). In cases like the one presented, where the slope is given, it serves as a starting point to write the equation of the line when combined with other data, such as a point on the line.

Y-Intercept

The y-intercept is a crucial element of the slope-intercept form equation \( y = mx + b \). It is the value of \( y \) where the line crosses the y-axis, represented by \( b \). To find the y-intercept when it is not given, you can use any known point on the line. In our problem, we used the point \((1, 3)\) to determine \( b \).Here's how you can find \( b \):1. Substitute the known slope and the coordinates of the given point into the equation.2. Solve for \( b \) using the equation \( y = mx + b \).In the example, after substituting \( x = 1 \) and \( y = 3 \), we calculated \( b = 9 \). This value represents the y-intercept, showing us exactly where the line will intersect the y-axis on a graph. Understanding how to find the y-intercept allows for a complete definition of a line in slope-intercept form.