Problem 36 - College Algebra

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(x\) -intercept at (-5,0) and \(y\) -intercept at (0,4)

Step-by-Step Solution

Answer

The linear equation is \(y = \frac{4}{5}x + 4\).

Step 1: Identify the Intercepts

The problem provides us with the \(x\)-intercept \((-5,0)\) and the \(y\)-intercept \((0,4)\). From this information, we know the line passes through both intercept points.

Step 2: Determine the Slope

Using the intercepts \((-5,0)\) and \((0,4)\), we can calculate the slope \(m\) of the line. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points, we get \(m = \frac{4 - 0}{0 - (-5)} = \frac{4}{5}\).

Step 3: Write the Linear Equation

With the slope \(m = \frac{4}{5}\) and the \(y\)-intercept of \(4\), we can write the equation of the line in the slope-intercept form, \(y = mx + b\), where \(b\) is the \(y\)-intercept. Thus, the equation becomes \(y = \frac{4}{5}x + 4\).

Step 4: Verification

To ensure our linear equation is correct, we can check if both intercepts satisfy the equation \(y = \frac{4}{5}x + 4\). For \(x = -5\), \(y = 0\), which satisfies \(y = \frac{4}{5}(-5) + 4 = 0\). For \(x = 0\), \(y = 4\), which satisfies \(y = \frac{4}{5}(0) + 4 = 4\). Both checks confirm the equation is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used and recognizable formats for writing the equation of a line. It is expressed as:
\[ y = mx + b \]
where:

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

This form is particularly useful because it allows you to easily identify both the slope and the y-intercept of a line just by looking at the equation. Knowing how to convert different forms of a line equation into the slope-intercept form will be immensely helpful for graphing and understanding line properties.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means that at the x-intercept, the y-value is always 0. To find the x-intercept from an equation, you can set \(y = 0\) and solve for \(x\).
In our problem, the x-intercept is given as the point \((-5, 0)\), indicating that the line crosses the x-axis at \(x = -5\).
Understanding x-intercepts is crucial because it informs you of where the graph will intersect the x-axis. In practical terms, for a physical system, the x-intercept might represent a point in time or space where a certain value reduces to zero.

Y-Intercept

The y-intercept is where the line meets the y-axis. At the y-intercept, the x-value is always 0. This point is often denoted by \((0, b)\) where \(b\) represents the intercept's y-value. Finding it involves setting \(x = 0\) and solving for \(y\).
In our exercise, the y-intercept is given as the point \((0, 4)\), showing that the line crosses the y-axis at \(y = 4\).
The y-intercept is easy to locate on a graph and plays a pivotal role in determining the vertical starting point of a line in the Cartesian plane when using the slope-intercept form equation.

Slope Calculation

Calculating the slope is crucial for understanding how steep a line is and in which direction it moves. The slope, denoted as \(m\), can be found using two points on the line by applying the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our example, we use the points \((-5,0)\) and \((0,4)\) to calculate the slope:

• Substitute the coordinates into the formula:\[ m = \frac{4 - 0}{0 - (-5)} = \frac{4}{5} \]

The positive slope \(\frac{4}{5}\) indicates that for each unit the line moves horizontally to the right, it moves \(\frac{4}{5}\) units up vertically. This gradient illustrates the line's incline over its run and is a foundational component in the slope-intercept equation format.