Problem 41 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -10 y=-12 x+1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 12/10 and the y-intercept is -1/10.

Step 1: Rearrange the given equation to the slope-intercept form

First, we'll divide both sides of the equation by -10 to isolate y: $$ -10 y = -12 x + 1 \implies y = \frac{-12}{-10} x + \frac{1}{-10} $$

Step 2: Identify the slope and y-intercept

Now that we have the equation in the form \(y= mx+b\), we can identify the slope and y-intercept: $$ y = \frac{12}{10} x - \frac{1}{10} $$ So, the slope \(m = \frac{12}{10}\), and the y-intercept \(b = -\frac{1}{10}\). Thus, the slope of the line is \(\frac{12}{10}\) and the y-intercept is \(-\frac{1}{10}\).

Key Concepts

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is one of the easiest ways to write a line equation. It is generally expressed as \( y = mx + b \). This form immediately tells you two crucial pieces of information about the line:

• The slope \( m \)
• The y-intercept \( b \)

The slope \( m \) represents how steep the line is, while the y-intercept \( b \) indicates where the line crosses the y-axis. Having the equation in this form is beneficial because it provides a quick way to sketch the graph or understand the characteristics of a line.
In our example, rearranging the equation \(-10y = -12x + 1\) into \(y = \frac{12}{10}x - \frac{1}{10}\) helps us swiftly identify these key features.

Decoding the Slope

The slope of a line is essentially a measure of its steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.This is why you often hear that slope equals \( \frac{\text{rise}}{\text{run}} \).
In mathematical terms, the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]In the slope-intercept form \( y = mx + b \), the slope \( m \) directly tells us:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it goes from left to right.

In our specific example, the slope \( \frac{12}{10} \) or \( 1.2 \) indicates a relatively gentle upward incline.

Significance of the Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It specifically occurs at the point where the value of \( x \) is zero.
In the context of the slope-intercept form equation \( y = mx + b \), the \( b \) value represents the y-intercept: the y-coordinate of the point where the line meets the y-axis.
If you picture the line on a graph, the y-intercept is a fundamental starting point for drawing your line. Using it, along with the slope, you can quickly plot a whole line.
In our equation \( y = \frac{12}{10}x - \frac{1}{10} \), the y-intercept is \( -\frac{1}{10} \), which is quite close to zero, indicating the line crosses the y-axis just below the origin.