Problem 55 - Precalculus with Limits

Question

In Exercises 51-64, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope \(m\). Sketch the line. \((4, 0)\), \(m = -\frac{1}{3}\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line that passes through the point (4, 0) and has a slope of -1/3 is \(y = -\frac{1}{3}x + \frac{4}{3}\).

Step 1: Identify the Given Values

Determine the given values in the problem. The point in the problem is (4, 0) and the slope is -1/3.

Step 2: Substitution into the Equation

Substitute the given values into the equation of the line in the slope-intercept form. The equation turns into \(0 = (-1/3)*4 + c\). This equation will help us find the y-intercept (c) of the line.

Step 3: Solve for the Y-Intercept

Solve the equation for \(c\). Adding 4/3 to both sides, we would get \(c = 4/3\). Thus, the y-intercept of the line is 4/3.

Step 4: Formulate the Line Equation

Substitute the slope (m) and the y-intercept (c) back into the equation of the line. This gives \(y = -\frac{1}{3}x + \frac{4}{3}\). This is the slope-intercept form of the line that passes through the point (4, 0) and has a slope of -1/3.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and represent relationships where the change in one variable is consistent with the change in another. In its simplest form, a linear equation is represented as
\(y = mx + b\),
where \(m\) is the slope of the line and \(b\) is the y-intercept. The beauty of linear equations lies in their simplicity and the straightforward nature of their graphs: straight lines.

When dealing with linear equations, it's crucial to recognize that the equation corresponds to a set of points on a coordinate plane that, when connected, form a straight line. This definitively consistent behavior allows us to predict and understand the relationship between variables efficiently.

Slope of a Line

Understanding the Slope

The slope of a line is a measure of its steepness or gradient. Mathematically, it's denoted as \(m\) and defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line, commonly expressed as
\(m = \frac{{\text{{rise}}}}{{\text{{run}}}}\).

Positive slopes ascend from left to right, indicating that as \(x\) increases, \(y\) also increases. Conversely, a negative slope descends from left to right, meaning that as \(x\) increases, \(y\) decreases. A slope of zero indicates a horizontal line, and an undefined or infinite slope corresponds to a vertical line. Slopes are central to understanding how variables in a linear equation interact with one another.

Y-Intercept

Finding the Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis of the coordinate plane. It is represented by the \(y\) in the equation
\(y = mx + b\),
where \(b\) is the actual y-intercept value. This intercept is a key component of the slope-intercept form and represents the value of \(y\) when \(x\) is zero.

Understanding the y-intercept can give great insight into the context of a problem—knowing where a line intersects the y-axis can help in graphing the equation or understanding the starting point of a scenario being modeled by a linear equation. Remember: the y-intercept is a fixed point, regardless of the slope, which provides an anchor point from which the line will rise or fall.