Problem 70 - Calculus Volume 1

Question

Write the equation of the line satisfying the given conditions in slope- intercept form. Slope \(=\frac{2}{5}, x\) -intercept \(=8\)

Step-by-Step Solution

Answer

Equation: \( y = \frac{2}{5}x - \frac{16}{5} \)

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Use Slope and x-Intercept to Find y-Intercept

We know the slope \( m = \frac{2}{5} \) and the x-intercept is 8. The x-intercept is the point where the line crosses the x-axis, meaning the line passes through the point \((8, 0)\). Substitute these into the equation to find \( b \).

Step 3: Substitute Values into the Equation

Substitute the point \((8, 0)\) into the equation \( y = mx + b \): \[ 0 = \frac{2}{5} \times 8 + b \].

Step 4: Solve for y-Intercept

Continuing from the substitution, calculate to find \( b \): \[ 0 = \frac{16}{5} + b \] Subtract \( \frac{16}{5} \) from both sides: \[ b = -\frac{16}{5} \].

Step 5: Write the Final Equation

Now that we have both the slope and y-intercept, write the final equation: \[ y = \frac{2}{5}x - \frac{16}{5} \].

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra and geometry. One of the most common forms to express a line's equation is the slope-intercept form, represented by \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept.

This form is particularly useful because it gives us immediate insights into the line's characteristics:

• The slope \( m \) tells us how steep the line is, describing the line's tendency to rise or fall.
• The y-intercept \( b \) indicates where the line crosses the y-axis.

In the given problem, we aim to express the line with a slope of \( \frac{2}{5} \) and an x-intercept of 8 in slope-intercept form. Using known values, we construct the equation by inserting the slope \( m = \frac{2}{5} \) and solve for the y-intercept \( b \). The final linear equation takes on the form \( y = \frac{2}{5}x - \frac{16}{5} \), giving a complete view of the line in a straightforward equation.

Y-Intercept

The y-intercept is a crucial point where a line intersects the y-axis. This happens where the value of x is zero. The y-intercept provides valuable information about the line’s position relative to the coordinate grid.

In the slope-intercept formula \( y = mx + b \), the term \( b \) represents the y-intercept. It shows us the height at which the line crosses the vertical axis. When given other conditions like an x-intercept and slope, as in our problem, it becomes possible to determine the y-intercept.

We start by using the x-intercept and substituting into the slope-intercept formula. Here, we substitute point \((8, 0)\) into the equation, use the known slope, and solve for \( b \). This calculation gives \( b = -\frac{16}{5} \), illustrating the point where the line crosses the y-axis.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this intercept, the y-coordinate is zero since the line is at the horizontal level of the axis. Knowing the x-intercept is extremely beneficial in finding the full equation of a line.

For any line given in slope-intercept form \( y = mx + b \), the x-intercept can be found by setting \( y = 0 \) and solving for \( x \). However, in our problem, we are provided with the x-intercept first, which is located at \((8, 0)\). This means when the x-value is 8, the y-value is zero. Using this information, along with the slope \( m = \frac{2}{5} \), allows us to back-calculate the y-intercept \( b \) for the equation.

Ultimately, the role of the x-intercept in this context is to help pinpoint where on the graph the line starts to ascend or descend, assisting in the verification and completion of the line's equation.