Problem 9 - 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: Identify the Equation

The given equation is in the general form \(7y + 12x - 2 = 0\). Our goal is to convert this into the slope-intercept form of a line, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Rearrange the Equation

We need to solve for \(y\) to put the equation in slope-intercept form. Start by isolating the \(7y\) term: \(7y = -12x + 2\).

Step 3: Solve for y

Divide every term by 7 to solve for \(y\): \(y = -\frac{12}{7}x + \frac{2}{7}\). Now the equation is in the form \(y = mx + b\).

Step 4: Identify the Slope and Y-Intercept

From the equation \(y = -\frac{12}{7}x + \frac{2}{7}\), identify the slope (\(m\)) and the y-intercept (\(b\)). 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 an incredibly useful way to express the relationship between two variables, typically denoted as \(x\) and \(y\). This form is given by the equation \(y = mx + b\). The best part about using the slope-intercept form is how neatly it displays the two most important features of a linear equation: the slope, represented by \(m\), and the y-intercept, represented by \(b\).

When a linear equation is in this form, you can easily identify the behavior of the line just by looking at \(m\) and \(b\). The process of converting a general form equation into the slope-intercept form involves solving it for \(y\), which typically involves rearranging and simplifying.

Once in slope-intercept form, interpreting a line becomes straightforward. You can instantly know how steep the line is and where it crosses the y-axis, giving you a quick sense of the line's overall path.

Slope

The slope of a line, denoted as \(m\), is one of the defining characteristics of a linear equation. It essentially measures the tilt or steepness of the line on a graph. In the slope-intercept form \(y = mx + b\), the slope \(m\) is the coefficient of \(x\).

A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. In our given example, the equation \(y = -\frac{12}{7}x + \frac{2}{7}\), the slope is \(-\frac{12}{7}\). This implies that for every increase of 7 units in \(x\), \(y\) decreases by 12 units.

• If the slope is zero, the line is perfectly horizontal.
• If the slope is undefined (such as when dividing by zero), the line is vertical.

Understanding slope is key to predicting how changes in one variable affect the other, making it a central concept in both algebra and calculus.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. Represented by \(b\) in the equation \(y = mx + b\), this value indicates what \(y\) is when \(x\) equals zero. In our case, with equations like \(y = -\frac{12}{7}x + \frac{2}{7}\), the y-intercept \(b\) is \(\frac{2}{7}\).

The beauty of the y-intercept in the graph is that it provides a visual marker for the line's starting point on the vertical axis. Knowing the y-intercept allows you to quickly plot the line on a graph by marking this point and using the slope to determine the line's direction.

• If \(b\) is positive, the intercept is above the origin.
• If \(b\) is zero, the line crosses the origin.
• If \(b\) is negative, the line starts below the origin.

Recognizing the y-intercept provides a straightforward way to begin understanding and sketching a line's graph efficiently. This concept is especially useful for quickly assessing linear relationships.