Problem 38 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(=4\) and \(y\) -intercept \(=-2\)

Step-by-Step Solution

Answer

The equation for the line is \(y = 0.5x - 2\)

Step 1: Determine the Slope

The slope, \(m\), is the change in \(y\) divided by the change in \(x\). In this case, the \(x\)-intercept is 4 and the \(y\)-intercept is -2. Thus, the slope is \(\frac{-2 - 0}{0 - 4} = 0.5\).

Step 2: Write in Point-Slope Form

Using the point-slope form equation, \(y - y1 = m(x - x1)\), replace \(m\) with the calculated slope and \((x1, y1)\) with either of the intercepts. Let's use the \(x\)-intercept (4, 0). Hence, the equation will be \(y - 0 = 0.5(x - 4)\). Simplifying this gives \(y = 0.5x - 2\).

Step 3: Write in Slope-Intercept Form

The slope-intercept form of a line's equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. The equation derived from the last step \(y = 0.5x - 2\) is already in the slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common forms used in mathematics. It is represented as \(y = mx + b\), where:

• \(m\) is the slope of the line, which measures how steep the line is.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is particularly useful because it allows you to quickly determine both the slope and the y-intercept of a line. You can easily sketch the graph of a line if you know these two features. When writing an equation in this form, you can see at a glance how the line behaves as it crosses the y-axis and what direction it moves.

X-Intercept

The x-intercept of a line is where the line crosses the x-axis. At this point, the y-value is zero, meaning the line touches the horizontal axis without reaching a vertical displacement. To find the x-intercept in a linear equation:

• Set \(y\) to zero in the equation.
• Solve for \(x\).

In this specific exercise, the x-intercept is given as 4. This tells you the line hits the x-axis at the point (4,0). X-intercepts reveal important information about a graph's behavior, such as where it starts or ends on a plot when moving left to right.

Y-Intercept

The y-intercept is where the line crosses the y-axis, and at this intercept, the x-value is zero. This point defines a critical feature of linear equations because it shows where the line meets the y-axis when it stretches across a graph. Finding the y-intercept involves:

• Setting \(x\) to zero in the equation.
• Simplifying to get the y-value.

In the exercise example, the y-intercept is -2. This point is represented as (0,-2) on a graph. Understanding the y-intercept is essential for drawing graphs and understanding the initial value or starting point of a function.

Linear Equations

Linear equations describe a straight line on a graph. They have various forms, including point-slope and slope-intercept, each useful for different applications. The general characteristics of linear equations are:

• They can be written in the form \(ax + by = c\) or \(y = mx + b\).
• The graph of a linear equation is always a straight line.
• They have constant slopes, meaning the rate of change is steady.

Understanding linear equations allows you to analyze and interpret graphs easily. You can determine intersection points, calculate slopes, and understand linear relationships between variables. Mastering this concept is crucial in algebra and pre-calculus.