Problem 40 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. With slope \(\frac{5}{7},\) through (0,-3)

Step-by-Step Solution

Answer

The equation is \( y = \frac{5}{7}x - 3 \).

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept of the line. We are given the slope \( m = \frac{5}{7} \) and a point on the line which allows us to find \( b \).

Step 2: Identifying the Y-Intercept

Since the point (0,-3) lies on the line, the y-coordinate of this point is the y-intercept. Therefore, we have \( b = -3 \).

Step 3: Writing the Equation

Substitute the values of slope \( m = \frac{5}{7} \) and the y-intercept \( b = -3 \) into the slope-intercept form equation \( y = mx + b \). This gives us \( y = \frac{5}{7}x - 3 \).

Step 4: Resulting Equation

The equation of the line with slope \( \frac{5}{7} \) passing through the point (0, -3) is \( y = \frac{5}{7}x - 3 \).

Key Concepts

Slope

The slope of a line is a measure that tells us how steep the line is. In the standard form of a linear equation, the slope is represented by the letter \( m \). Essentially, the slope is calculated as the "rise over run," meaning how much the line goes up (or down) for every step it takes sideways. This can be mathematically expressed as:

• Rise: Change in y-values (vertical change)
• Run: Change in x-values (horizontal change)

If you imagine climbing a hill, a steeper hill has a larger slope, while a gentle slope has a smaller number. A horizontal line, which has no rise, has a slope of zero. Likewise, a vertical line, often described as having an undefined slope because there's no horizontal movement, can be thought of as being infinitely steep.

In the given example, the slope is \( \frac{5}{7} \), meaning for every 7 units you move to the right on the x-axis, the line moves 5 units up along the y-axis.

Y-Intercept

The y-intercept is where the line crosses the y-axis on a graph. It is a crucial part of a linear equation in slope-intercept form (\( y = mx + b \)), because it represents the point where \( x = 0 \).

In simpler terms, it's the value of \( y \) when the line meets the y-axis. The y-intercept is particularly useful because it provides a starting point for graphing the line before using the slope to find other points.

• Intercept: The value of \( y \) when \( x=0 \)

Given that the point \((0, -3)\) is on the line, it helps us immediately identify the y-intercept, which is \( b = -3 \). This tells us that the line crosses the y-axis 3 units below the origin.

Linear Equation

A linear equation describes a straight line on a graph and is one of the simplest forms of an equation in mathematics. Every linear equation can be represented in slope-intercept form: \( y = mx + b \). This format succinctly includes all the essential information needed to graph the line:

• Slope \( m \): Describes the angle or steepness of the line
• Y-Intercept \( b \): Shows where the line crosses the y-axis

The beauty of a linear equation lies in its ability to model constant rates of change. This means wherever you are on the line, moving sideways by the same amount changes the y-value by a constant factor, making it predictable and useful in real-world applications like predicting income over time, calculating speed, or determining price trends.

For the provided problem, using the known slope \( \frac{5}{7} \) and y-intercept \( -3 \), we substitute these values into the slope-intercept form to find our linear equation: \( y = \frac{5}{7}x - 3 \). This equation effectively models a line with all its unique characteristics defined by the specific numerical values of slope and intercept.