Problem 70 - Calculus Volume 1

Question

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form. $$ =\frac{2}{5}, x \text { -intercept }=8 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{2}{5}x - \frac{16}{5} \).

Step 1: Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We are given the slope as \( \frac{2}{5} \) and need to find the y-intercept \( b \).

Step 2: Identify the X-Intercept

The x-intercept is the point where the line crosses the x-axis, which means the y-value is zero. We are given that the x-intercept is 8, so this corresponds to the point \((8, 0)\) on the graph.

Step 3: Use the Point-Slope Relationship

Using the point-slope form of the equation \( y = mx + b \), and knowing that at the x-intercept, the y-value is 0, substitute the x-intercept value and the slope into the equation:\[0 = \frac{2}{5}(8)+b\]

Step 4: Solve for the Y-Intercept \( b \)

Substitute \( x = 8 \) into the equation and solve for \( b \):\[0 = \frac{2}{5}(8) + b\]\[0 = \frac{16}{5} + b\]Subtract \( \frac{16}{5} \) from both sides:\[b = -\frac{16}{5}\]

Step 5: Write the Final Equation

Now that we have both the slope \( m = \frac{2}{5} \) and the y-intercept \( b = -\frac{16}{5} \), we can write the equation of the line in slope-intercept form:\[y = \frac{2}{5}x - \frac{16}{5}\]

Key Concepts

Linear Equations

A linear equation is essentially any equation that can be represented as a straight line on a graph. The most popular form of a linear equation is the slope-intercept form, which takes the format of \( y = mx + b \). Here, \( m \) represents the slope of the line \[ m = \frac{rise}{run} \]and \( b \) represents the y-intercept, or where the line crosses the y-axis.
The slope tells you how steep the line is, and the y-intercept tells you the exact value at which the line crosses the y-axis. This form allows you to quickly graph a line and understand its key properties.
When you hear 'linear equations,' picture perfect straight lines that can forever extend in both directions on a graph.

X-Intercept

The x-intercept of a line is the point at which the line crosses the x-axis. At this point, the value of \( y \) is always zero. It’s identified by the coordinate \((x, 0)\). Finding the x-intercept is immensely helpful for graphing, as it serves as a landmark.

• The x-intercept can be calculated by setting \( y = 0 \) in the equation and solving for \( x \).

The importance of the x-intercept is evident in tasks like solving real-world problems where you need to find out when a quantity becomes zero. Consider it as a starting point to map a line on the graph accurately.

Y-Intercept

The y-intercept is where a line meets the y-axis, which means at this crossing point, the x-value is zero. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept. It denotes the starting point of the line when plotted on a graph.
To find the y-intercept, substitute \( x = 0 \) into the equation and solve for \( y \).

• It's imperative for understanding how a line behaves concerning its height on the graph.

Knowing the y-intercept helps in drafting the line quickly and is particularly useful in evaluating the relative position of two lines. Always remember, the y-intercept is your line's anchor on the y-axis!

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. Here, we find the equation of a line using certain given conditions. The process begins by identifying what you know, such as the slope or an intercept, and then finding the y-intercept to write a complete equation.

• First, recognize which values you have: slope (\( m \)) or intercepts (x or y).
• Use these to plug into the formula \( y = mx + b \) for slope-intercept form.

In our problem with an x-intercept of 8 and a slope of \( \frac{2}{5} \), we solved for \( b \) by setting up the equation based on \( 0 = \frac{2}{5}(8) + b \). This led us to discover \( b = -\frac{16}{5} \). Solve equations step-by-step, and you'll unravel the mystery of the line piece by piece.