Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-38, find the slope and \(y\)-intercept of each line. \(6-2 y=10 x-2\)

Step-by-Step Solution

Answer

The slope is 5, and the y-intercept is -4.

Step 1: Rearrange the Equation

We start with the equation given: \( 6 - 2y = 10x - 2 \). Our goal is to express it in the slope-intercept form \( y = mx + b \). First, we'll isolate terms involving \( y \). Add \( 2y \) to both sides to get: \( 6 = 10x - 2 + 2y \).

Step 2: Simplify the Equation

Now, we add \( 2 \) to both sides to isolate terms involving \( x \) and constant terms on one side: \( 8 = 10x + 2y \).

Step 3: Solve for y

Reorder the equation as \( 2y = 10x - 8 \) by subtracting \( 8 \) from both sides. Divide every term by 2 to express \( y \): \( y = 5x - 4 \).

Step 4: Identify the Slope and y-intercept

Now that we have the equation \( y = 5x - 4 \), we see that it's in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Thus, the slope \( m \) is \( 5 \), and the \( y \)-intercept \( b \) is \( -4 \).

Key Concepts

Linear Equations

Linear equations are foundational components in algebra. These equations describe straight lines when graphed on a coordinate plane. The standard form of a linear equation is given by \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. However, in many scenarios, it's more useful to transform these equations into the slope-intercept form, which is \( y = mx + b \).

• \( m \) represents the slope of the line, indicating the steepness and direction.
• \( b \) is the y-intercept, highlighting where the line crosses the y-axis.

Understanding the transformation from the standard form to the slope-intercept form is crucial. This manipulation reveals critical features of the linear relationship, such as slope and intercept, making it easier to interpret graphically.

Slope Calculation

The slope of a line tells us how sharply a line rises or falls as you move from left to right across the graph. It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate. In our solution, the equation was first adjusted to the form \( y = mx + b \) to easily identify \( m \) (the slope). Here's how we transformed the equation:

• Initial equation: \( 6 - 2y = 10x - 2 \)
• Rearranged to isolate \( y \): \( 2y = 10x - 8 \)
• Simplified to slope-intercept: \( y = 5x - 4 \)

In this situation, the slope \( m = 5 \). This value signifies that for every unit increase in x, the value of y increases by 5 units. Thus, the line rises steeply, moving upwards to the right.

Intercept Identification

The y-intercept of a linear equation in slope-intercept form \( y = mx + b \) is the value \( b \). It specifies the point at which the line crosses the y-axis. This is a critical part because it provides a starting point for graphing the line.In the adjusted equation \( y = 5x - 4 \), the intercept \( b = -4 \) reveals that the line crosses the y-axis at the point (0, -4). This means that when \( x = 0 \), \( y \) will be \(-4\). Knowing the y-intercept helps in quickly sketching the graph of the line since you have a specific point to start from. By clearly identifying the y-intercept and combining it with the slope, graphing becomes straightforward.