Problem 17 - Calculus Single Variable

Question

Write the slope-intercept equation of the line determined by the given data. Slope \(3, x\) -intercept -4

Step-by-Step Solution

Answer

The equation is \(y = 3x + 12\).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to find \(y = mx + b\) given the slope and the \(x\)-intercept.

Step 2: Find the Y-Intercept Using the X-Intercept

The \(x\)-intercept is the point where the line crosses the \(x\)-axis, which means \(y = 0\). We are given the \(x\)-intercept as -4. Substitute \(y = 0\) and \(x = -4\) into the equation \(y = mx + b\) to find \(b\): \(0 = 3(-4) + b\).

Step 3: Solve for the Y-Intercept

Solving the equation from Step 2, \(0 = -12 + b\), gives \(b = 12\). Now we have the \(y\)-intercept.

Step 4: Substitute Values into Slope-Intercept Formula

We have a slope \(m = 3\) and a y-intercept \(b = 12\). Substitute these values into the slope-intercept equation: \(y = 3x + 12\). Therefore, the equation of the line is \(y = 3x + 12\).

Key Concepts

Equation of a Line

An equation of a line represents a straight line on a graph, which is why it is often referred to as a linear equation. The standard format for a line's equation in the slope-intercept form is \(y = mx + b\). This equation provides a clear way to understand the characteristics of the line on a coordinate plane.
In this form:

• \(y\) represents the dependent variable or the vertical position on the graph.
• \(x\) represents the independent variable or the horizontal position on the graph.
• \(m\) signifies the slope of the line.
• \(b\) indicates the point where the line crosses the y-axis, known as the y-intercept.

The equation tells us not only where the line is positioned but also how it changes across the graph. This information is vital for graphing linear relationships or understanding how changes in one variable affect another.

Y-Intercept

The y-intercept is a critical part of understanding a line's position on a graph. The y-intercept is the point where the line crosses the y-axis. In mathematical terms, it is the value of \(y\) when \(x = 0\). This helps us determine the starting point of the line on the vertical axis.
To find the y-intercept, you use the slope-intercept form of the line, \(y = mx + b\).

• The equation is written such that \(b\) is already the y-intercept.
• If you're given a slope and another point on the line, you can substitute those values into the equation to solve for \(b\).

In the given exercise, we found the y-intercept using the x-intercept and the slope. By substituting the \(x\)-intercept into the formula \(y = mx + b\), we calculated \(b = 12\), telling us where the line hits the y-axis.

X-Intercept

Understanding the x-intercept provides further insight into a line's graph. The x-intercept is where the line crosses the x-axis, meaning \(y = 0\) at this point. It indicates the input value \(x\) for which the resulting \(y\) coordinate is zero.
To find the x-intercept:

• Set \(y = 0\) in the equation of the line.
• Solve for \(x\).

In the exercise, the x-intercept was provided as \(-4\). Using this information along with the slope, we effectively determined the y-intercept by substituting \((x, y) = (-4, 0)\) into the equation, leading us to the full line equation.

Linear Equations

Linear equations are foundational in algebra, describing relationships that graph as straight lines. They show a constant rate of change, highlighted through their slope. Linear equations can be used in various forms, such as standard form and slope-intercept form. The slope-intercept form, \(y = mx + b\), is particularly useful for quickly identifying both the slope and the y-intercept.
These equations are ubiquitous in the real world and can model many types of natural and scientific phenomena, such as:

• Predicting trends in business through financial forecasting
• Determining speed in basic physics calculations
• Comparing costs in economic models

Understanding and applying linear equations, such as we did in the exercise by calculating the slope and intercepts, equips students with tools for problem-solving in both academic settings and everyday life.