Problem 46 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(1,\) through (-7,9)Slope \(5,\) through (6,-8)

Step-by-Step Solution

Answer

The equations of the lines are \( y = x + 16 \) and \( y = 5x - 38 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line's equation is given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our goal is to find the value of \( b \) using the given slope and a point on the line.

Step 2: Equation for Line with Slope 1

For the line with slope 1, which passes through the point \((-7, 9)\), we substitute into the slope-intercept form:\[9 = 1(-7) + b\]Solve for \( b \):\[9 = -7 + b \Rightarrow b = 9 + 7 = 16\]Thus, the equation is \( y = x + 16 \).

Step 3: Equation for Line with Slope 5

For the line with slope 5, which passes through the point \((6, -8)\), we substitute into the slope-intercept form:\[-8 = 5(6) + b\]Solve for \( b \):\[-8 = 30 + b \Rightarrow b = -8 - 30 = -38\]Thus, the equation is \( y = 5x - 38 \).

Key Concepts

Slope

The slope of a line is a measure of how steep the line is. It describes the direction and the incline of the line. Think of it as the "rise over run," meaning how much the line goes up or down (rise) for each step it takes sideways (run). For example, if the slope is 1, the line rises by 1 unit for every 1 unit it moves to the right. In mathematical terms, the slope is denoted by the letter \( m \) and is calculated by dividing the change in the \( y \)-coordinates by the change in the \( x \)-coordinates between two distinct points on the line:

• Slope, \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope means the line inclines upward as it moves along, while a negative slope means it declines. A slope of zero represents a horizontal line, and an undefined slope usually means a vertical line. Understanding the slope is crucial as it not only tells us how the line behaves but also serves as an important component in writing the equation of the line in slope-intercept form.

Slope-Intercept Form

The slope-intercept form is the most common way to express the equation of a line. It is written as:

• \( y = mx + b \)

In this equation format, \( m \) represents the slope and \( b \) represents the y-intercept.This form is particularly useful because it provides direct insights about the line:

• The slope (\( m \)) gives the steepness and direction of the line.
• The y-intercept (\( b \)) shows where the line crosses the \( y \)-axis.

To use the slope-intercept form, you’ll need to know the slope (\( m \)) and at least one point on the line to solve for \( b \), the y-intercept. For instance, if you have a slope of 1 and a point (-7, 9), you can plug these into the slope-intercept form:

• \( 9 = 1(-7) + b \)
• Solve for \( b \) to find \( b = 16 \), making the equation \( y = x + 16 \).

This mode of equation representation is helpful for quickly sketching the graph of a line because it tells you immediately the starting point on the \( y \)-axis and how the line moves as \( x \) increases or decreases.

Y-intercept

The y-intercept is a critical concept in understanding and graphing lines. It is the point where the line crosses the \( y \)-axis, and it is a direct representation of the constant term \( b \) in the slope-intercept form \( y = mx + b \).The y-intercept indicates the \( y \)-value at which the line intersects when the \( x \)-coordinate is zero, essentially showing where the line starts on the vertical axis. It is denoted by the point \( (0, b) \).For example, consider a line with the equation \( y = 5x - 38 \). Here, the y-intercept \( b \) is \(-38\), indicating that the line crosses the \( y \)-axis at the point \( (0, -38) \).Understanding the y-intercept is essential when attempting to graph a line because it provides a starting position from which the slope can guide the drawing of the line across the plane. It also helps solve real-world problems where initial values play a role, such as calculating starting costs before incremental expenses add up.