Problem 27 - Elementary Algebra

Question

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=-\frac{1}{6}\) and \(b=-4\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{1}{6}x - 4 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. We need to substitute the given values of \( m \) and \( b \) into this form to find the equation of the line.

Step 2: Substitute the Given Slope and Y-Intercept

We are given the slope \( m = -\frac{1}{6} \) and the y-intercept \( b = -4 \). Substitute these values into the formula: \[ y = -\frac{1}{6}x - 4 \].

Step 3: Verify the Equation

Verify that the equation \( y = -\frac{1}{6}x - 4 \) meets the given conditions by checking it includes the correct slope and y-intercept as provided.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, acting as the building blocks for understanding how relationships between variables are structured. At its core, a linear equation is an equation that forms a straight line when graphed on a coordinate plane. This can be represented in various forms, such as slope-intercept, point-slope, and standard form.

When we talk about linear equations, it's important to acknowledge the role of each component:

• The variable, usually denoted by \(x\), represents an unknown value that we aim to solve for.
• The coefficient of \(x\) determines how much influence \(x\) has on the equation.
• The constant term in the equation shifts the line vertically on the graph.

Linear equations always graph to a straight line, facilitating an easy and visual interpretation of data. Understanding these equations is crucial since they lay the groundwork for more complex mathematical concepts.

Slope

The slope of a line is a measure of its steepness and direction. It tells us how much the line slants and in which direction it moves. Mathematically, slope is defined as the ratio of the change in \(y\) (vertical change) to the change in \(x\) (horizontal change). In formula terms, this is often expressed as \(m = \frac{\Delta y}{\Delta x}\).

The slope has a few important characteristics:

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves across the graph.
• A zero slope results in a horizontal line, implying that \(y\) does not change as \(x\) varies.
• An undefined slope (division by zero) indicates a vertical line.

In the slope-intercept form of a linear equation, represented as \(y = mx + b\), \(m\) is the slope. For our example, the slope is given as \(-\frac{1}{6}\), which means the line is almost flat but descends slightly as you move along the \(x\)-axis.

Y-Intercept

The y-intercept plays a vital role in the slope-intercept form of a linear equation. This interception point is where the graph of a line crosses the \(y\)-axis. It provides a starting point for applying the slope to graph the line accurately.

The y-intercept is denoted by \(b\) in the equation \(y = mx + b\). When the \(y\)-intercept \(b\) is positive, the line crosses the \(y\)-axis above the origin. If \(b\) is negative, the line intersects below the origin. In the provided problem, the y-intercept is \(-4\), which means the line crosses the \(y\)-axis four units below the origin.

Understanding the y-intercept helps you quickly draw the line on a graph by starting at \(b\) on the \(y\)-axis and using the slope \(m\) to find other points on the line. Together, the slope and y-intercept describe the key characteristics of a linear equation in slope-intercept form.