Problem 1 - Applied Calculus

Question

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$ 7 y+12 x-2=0 $$

Step-by-Step Solution

Answer

Slope: \(-\frac{12}{7}\), Y-intercept: \(\frac{2}{7}\).

Step 1: Rearrange the Equation

Start by rearranging the equation in order to solve for \( y \). The original equation is \( 7y + 12x - 2 = 0 \). Move \( 12x \) and \( -2 \) to the other side of the equation.\[ 7y = -12x + 2 \]

Step 2: Solve for y

Divide every term by 7 to express \( y \) in terms of \( x \).\[ y = -\frac{12}{7}x + \frac{2}{7} \]

Step 3: Identify the Slope and Y-Intercept

The equation \( y = mx + b \) represents the slope-intercept form where \( m \) is the slope and \( b \) is the y-intercept. From the rearranged equation \( y = -\frac{12}{7}x + \frac{2}{7} \), the slope \( m \) is \( -\frac{12}{7} \) and the \( y \)-intercept \( b \) is \( \frac{2}{7} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to easily understand the relationship between the two variables. This form is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the \( y \)-intercept.

This format is particularly useful because it quickly reveals key information about the line. By simply looking at the equation, you can instantly see both the slope and where the line crosses the \( y \)-axis.

• The equation is simple to create from any linear equation once you isolate \( y \) on one side.
• It makes graphing straightforward because you can start plotting at the \( y \)-intercept and use the slope to find the next points.

Understanding this form helps in quickly analyzing linear relationships in various mathematical contexts.

Slope of a Line

The slope of a line is a crucial concept that indicates how steep the line is. It is usually represented by the letter \( m \) in equations. The slope is calculated as the ratio of the change in the \( y \)-value to the change in the \( x \)-value between two points on the line. Mathematically, it can be expressed as \( m = \frac{\Delta y}{\Delta x} \).

Here's what you need to know about the slope:

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• A slope of zero means the line is horizontal.
• An undefined slope indicates a vertical line.

In our example, the slope \( m \) is \(-\frac{12}{7}\), making the line fall as you go from left to right. This slope value tells us how much 'y' changes with every unit increase in 'x'. Knowing the slope helps in understanding the trend of the data or direction of the relationship shown by the line.

Y-Intercept

The \( y \)-intercept is another essential aspect of linear equations and usually symbolized as \( b \) in the slope-intercept form. It represents the point where the line crosses the \( y \)-axis. In simpler words, it's the \( y \)-value when \( x = 0 \).

Here are some important points about the \( y \)-intercept:

• It provides a starting point for graphing a line. You begin at \( b \) on the \( y \)-axis.
• It reflects the value of the dependent variable when the independent variable is zero.
• In the equation \( y = -\frac{12}{7}x + \frac{2}{7} \), the \( y \)-intercept is \( \frac{2}{7} \).

Understanding the \( y \)-intercept allows you to quickly get a sense of where the line sits in relation to the \( y \)-axis. Having both the slope and \( y \)-intercept gives you complete insight into the linear equation's graph and behavior.