Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \((-10,4)\) \(m=0\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 4\). This will be a horizontal line crossing the y-axis at \(y = 4\).

Step 1: Understand the information given

The given point is (-10,4) and slope is \(m=0\). The slope being 0 indicates a horizontal line.

Step 2: Substitute the given values into the line's formula

Substitute the given point and slope into \(y = mx + b\) to solve for \(b\). Here, \(x = -10, \quad y = 4, \quad m = 0\). So we get, \(4 = 0*(-10) + b\).

Step 3: Solve for b

Solving for \(b\) gives \(b = 4\). This tells us the point where the line cross the y axis.

Step 4: Equate the line normal form

Substitute \(m\) and \(b\) back into the line's formula to get the line equation \(y = 0*x + 4\), or in the simplified form \(y = 4\).

Step 5: Sketch the line

To sketch the line, first plot the point (-10,4), then draw a horizontal line crossing \(y = 4\) on the graph since the slope is zero.

Key Concepts

Understanding Slope-intercept Form

The slope-intercept form is a way to express the equation of a straight line. It is written as \(y = mx + b\). Using this formula, we can identify two critical characteristics of a line:

• The slope \(m\), which indicates the steepness or incline of the line.
• The y-intercept \(b\), which shows where the line crosses the y-axis.

If you have the slope and a point on the line, you can find the equation of that line. The slope measures the ratio of the change in y to the change in x, essentially how much \(y\) increases or decreases as you move along the line. If the slope is zero, as in our exercise, the line is perfectly horizontal, indicating no change in y as x changes. Therefore, the slope-intercept form simplifies to \(y = b\) for horizontal lines.

Identifying a Horizontal Line

A horizontal line is a straight line that goes from left to right, parallel to the x-axis. In terms of equations, a horizontal line has a slope \(m\) of zero since there is no vertical change regardless of the horizontal movement. This means in the slope-intercept form \(y = mx + b\), the value of \(m\) is zero.Because the slope is zero, the equation simplifies to \(y = b\). In this expression, \(b\) is where the line crosses the y-axis. Horizontal lines have the same y-value at every point along the line, creating a flat line. For instance, when you solve for equation in the problem \((y = 4)\), it means that every point on this line has a y-coordinate of 4. Horizontal lines are an excellent example of constancy in graphing.

Graphing Lines Made Easy

Graphing a line begins with understanding its slope and intercepts. For the case of horizontal lines:

• Identify the y-intercept (\(b\)), which will be a constant y-value.
• Since the slope \(m\) is zero, this line will not tilt up or down.

To graph a line like this:1. Plot the y-intercept on the y-axis. Using our example, plot the point \((0, 4)\).2. Draw a straight line parallel to the x-axis through this point.Graphing becomes straightforward when dealing with horizontal lines. You essentially draw a straight line that runs parallel to the x-axis, maintaining a constant y-value, making it simple and easy to visualize. It reinforces the idea that these lines have no change in their y-value, regardless of the x-coordinates.