Problem 13 - Calculus Single Variable

Question

Write the slope-intercept equation of the line determined by the given data. Slope \(-4, y\) -intercept 9

Step-by-Step Solution

Answer

The equation is \( y = -4x + 9 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the \( y \)-intercept.

Step 2: Substitute the Given Slope

We are given that the slope \( m \) of the line is \(-4\). Substitute \(-4\) for \( m \) in the slope-intercept form. This gives the equation: \( y = -4x + b \).

Step 3: Substitute the Given Y-Intercept

We are given that the \( y \)-intercept \( b \) is \( 9 \). Substitute \( 9 \) for \( b \) in the equation. This gives the final equation: \( y = -4x + 9 \).

Key Concepts

Understanding Linear Equations

A linear equation is an equation that graphs a straight line when plotted on a coordinate plane. The general form of a linear equation is given as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. The common forms make them easier to work with and understand how they represent a line. One of the most useful types of linear equations is the slope-intercept form, which provides clear information about the line's incline and where it crosses the y-axis.

The Role of Slope in Linear Equations

The slope is a measure of how steep a line is. In mathematical terms, it measures the change in the \( y \)-coordinate for a unit change in the \( x \)-coordinate. This is why it is often referred to as 'rise over run'. In the slope-intercept form \( y = mx + b \), the symbol \( m \) represents the slope. If the slope is positive, the line inclines upwards as it moves to the right; if negative, it declines.

• A slope of \( 0 \) indicates a horizontal, flat line.
• The greater the magnitude of the slope, the steeper the line.
• A slope of \(-4\), as in this example, means that for every 1 unit you move to the right, you move down 4 units.

Understanding the slope makes it easier to predict the trajectory of the line on a graph.

Deciphering the Y-Intercept

The \( y \)-intercept is the point where the line crosses the y-axis of the graph. It is represented by the symbol \( b \) in the slope-intercept equation \( y = mx + b \). The \( y \)-intercept tells you the starting point of the line when \( x = 0 \). This gives a precise location of the line within the coordinate plane.

• In our example, the \( y \)-intercept is \( 9 \). This means when \( x = 0 \), \( y = 9 \).
• The \( y \)-intercept is crucial because it anchors the line on the graph.
• It helps to quickly plot the line, as one point on the line is always known.

This anchor point provides a simple way to visualize and start drawing the line from the coordinate plane. Combined with the slope, it perfectly outlines the line's path.