Problem 11 - Calculus

Question

Determine the slope and the \(y\) -intercept of the line whose equation is given. $$12 x=6 y+4$$

Step-by-Step Solution

Answer

The slope is 2, and the y-intercept is \(-\frac{2}{3}\).

Step 1: Rearrange the Equation

First, rearrange the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The original equation is \( 12x = 6y + 4 \). To rearrange it, subtract 4 from both sides to get \( 12x - 4 = 6y \).

Step 2: Solve for y

Next, solve for \( y \) by dividing every term by 6 to isolate \( y \). The equation becomes \( y = 2x - \frac{2}{3} \). Now it's in the form \( y = mx + b \).

Step 3: Identify the Slope

Now that the equation is in the slope-intercept form \( y = 2x - \frac{2}{3} \), identify the slope \( m \) as the coefficient of \( x \), which is 2.

Step 4: Identify the Y-intercept

The y-intercept \( b \) is the constant term in the equation \( y = 2x - \frac{2}{3} \), which is \(-\frac{2}{3}\).

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is a widely used method in calculus to describe a straight line. It takes the form: \[y = mx + b \] where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

This format is particularly useful because it gives you immediate information about the line.

If the slope \( m \) is positive, the line rises as it moves from left to right. Conversely, if the slope is negative, the line falls. Working in the slope-intercept form makes it easier to graph a line or understand its properties, which is why rearranging an equation into this form is often the first step in solving linear equations.

Equation of a line

An equation of a line can be written in different forms, and each form provides unique insights about the line.

The general form is \[Ax + By = C\] where \( A \), \( B \), and \( C \) are constants. This form is less helpful for directly extracting the slope or the y-intercept.

By converting this to the slope-intercept form, linear characteristics become more explicit, making it easy to see:

• The slope, which dictates the line's direction and angle.
• The y-intercept, which signals where the line will meet the y-axis.

Knowing how to transform an equation into the slope-intercept form enhances your understanding of the line's behavior and supports better graphing capabilities.

Rearranging equations

Rearranging equations is a critical skill in mathematics, especially for solving and analyzing linear functions. The goal is to rewrite the equation to make one specific variable the subject, typically to clear up any confusion and simplify calculations.

For example, the equation given in the exercise: \[12x = 6y + 4\]can be rearranged to fit the familiar slope-intercept form \[y = mx + b\]Here's how you do it step-by-step:

• Subtract 4 from both sides to eliminate the constant on the right: \( 12x - 4 = 6y \).
• Divide every term by 6 to isolate \( y \): \( y = 2x - \frac{2}{3} \).

You'll then have the equation where the y-intercept and slope are clearly visible. By mastering these steps, you can simplify any line equation and gain intuitive insights about the line's properties.